Number question

[Jeffmaster223 24]

Hey guys, i'm back!

here's a tough question, and it involves math:

What number is divisible by 10 and below?

the number must be a whole number, and it must be the lowest # possible.

IT's SUPER HARD!!!

Please reply as quickly as possible

-Anim.

--try and find the answer!

  Edited April 19, 2011 by Animaniacs  

[Easy Bakes 253]

All of them?

you just do not get a whole number.

[Meg 368]

If I understand the question, it's not super hard. 2520.

You mean a number that is divisible by 10, 9, 8, 7, 6, 5, 4, 3, and 2?

take each number and multiply them together:

= 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2

but I'm guessing you want the smallest possible number so you need to factor them out and eliminate the duplicate factors:

= (5 * 2) * (3 * 3) * (2 * 2 * 2) * 7 * (3 * 2) * 5 * (2 * 2) * 3 * 2

= 5 * 2 * 3 * 3 * 2 * 2 * 2 * 7 * 3 * 2 * 5 * 2 * 2 * 3 * 2

= 5 * 2 * 3 * 3 * 2 * 2 * 2 * 7 * 3 * 2 * 5 * 2 * 2 * 3 * 2

= 5 * 2 * 3 * 3 * 2 * 2 * 7

= 2520

or did I not understand the question?

[A Nonny Moose 8,261]

Here is one for the numerologically minded.

Six is the first perfect number, being the sum of its factors (1,2,3). {Yes, 1 is always a factor}

What is the next one?

[Jeffmaster223 24]

MEG: i believe you do understand the question. I got 3628800,

but i don't know if that is the LCM of 1,2,3,4,5,6,7,8,9, and 10.

i got it by multipling all of the numbers together.

[Jeffmaster223 24]

A NONNY MOOSE: OOOOOOOO..... that's another good one.

i think it's...16.

[Duke87 1,070]

Nope, factors of 16 are 1, 2, 4, and 8. 1+2+4+8=15.

The answer is 28. Followed by 496 and 8,128. Next time try a question that it isn't ridiculously easy to just look up the answer to.

[Meg 368]

MEG: i believe you do understand the question. I got 3628800,

but i don't know if that is the LCM of 1,2,3,4,5,6,7,8,9, and 10.

i got it by multipling all of the numbers together.

Multiplying all of the numbers together will work but it usually isn't the least common multiple.

If you didn't like the method I used a few posts ago, try this one:

Take it one number at a time, checking if you need to add in an additional factor.

1 - is always there so skip that.

2 - We need a 2 so include it. (so far: 2)

3 - We need a 3 so include it. (so far: 2 * 3)

4 - We need (2 * 2). We only have one 2 so we need to include another one. (so far: 2 * 3 * 2)

5 - We need a 5 so include it. (so far: 2 * 3 * 2 * 5)

6 - We need (3 * 2). We already have that. (so far: 2 * 3 * 2 * 5)

7 - We need 7 so include it. (so far: 2 * 3 * 2 * 5 * 7)

8 - We need (2 * 2 * 2). We only have (2 * 2) so we need another 2. (so far: 2 * 3 * 2 * 5 * 7 * 2)

9 - We need (3 * 3) We only have one 3 so we need another 3. (so far: 2 * 3 * 2 * 5 * 7 * 2 * 3)

10 - We need (5 * 2). We already have that. (so far: 2 * 3 * 2 * 5 * 7 * 2 * 3)

2 * 3 * 2 * 5 * 7 * 2 * 3 = 2520

[Duke87 1,070]

That is certainly an interesting method you've got there. They never taught us anything like that. Our method of finding LCMs was guess and check. Or, since we only ever really used this in the context of adding and subtracting fractions, simply multiplying the numbers together sufficed - all you need is a common multiple, doesn't have to be the least common multiple.

Of course, it's easy to see why they didn't teach us this. They taught us what factors were very early on but they didn't teach us the concept of factoring things out until we were doing it in algebra.

One key thing that I think has changed in the philosophy of US public education between when you were in school, Meg, and when I was in school, is that the bar is set lower than it used to be. Used to be that you'd set the bar high, push students, and try and get as many of them over the high bar as you could. Nowadays, in the era of standardized testing and political correctness, the bar is set externally, and the name of the game isn't "make everyone the best they can be", it's "make everyone good enough". And altogether too much concern is given to the matter of whether boys and girls are performing equally and whether various ethinc groups are performing equally. All to the effect that education gets reduced to the lowest common denominator (aha!), and you have a general dumbing down of everything - because if you make it too un-dumbed down you start to expose people's flaws and inequalities, and good heavens we can't make it look like people are flawed and unequal! Everyone is supposed to have the same potential! The handicapper general at work.

[Meg 368]

One key thing that I think has changed in the philosophy of US public education between when you were in school, Meg, and when I was in school, is that the bar is set lower than it used to be.

I believe you are right. Standardized testing isn't helping anyone. and "No Child Left Behind" is a disaster for special ed students (as we used to call it in my day).

So, they don't teach this stuff in 4th grade anymore? That's when I learned it.

[A Nonny Moose 8,261]

Well, of course you could look it up. Now, what if there was no electricity and you had to work it out with a pencil, and without a reference library?

One of the biggest problems we have today is that students in school are dependent on the net and/or calculators. How many of them can do long division with a pencil and paper? You should try it sometime with Roman Numerals. In the Roman times, you had to be a mathematical genius, usually an idiot savant, to do that.

Do any schools below the post-secondary level teach Euclidean Geometry any more? It is a great discipline and teaches logic and self-reliance.

One college course I took covered the Erlangen Program (a survey of non-Euclidean geometries). The prof started by stating that we would consider the first five books of Euclid as axioms. We wound up deriving the General Theory of Relativity by approaching it through topology in a purely synthetic geometry without tensor calculus.

Another surprising relationship that many students have problems with is the one that combines all of mathematics in one equation.

e^iπ-1 = 0

[astronelson 339]

On the topic of Lowest Common Multiples (LCMs):

The LCM of all positive integers less than a given integer is the product of the highest powers of all the primes less than the given integer such that the power is also less than the given integer.

So for 10, these are 2³x3²x5¹x7¹=2520. You may notice this is the same as for 9.

I do not know what the LCM of all positive integers less than or equal to 100 is as a single integer, but I can tell you it is 2⁶x3⁴x5²x7²x11x13x17x19x23x29x31x37x41x43x47x53x59x61x67x71x73x79x83x89x97.

(In case you were wondering, I later calculated this. It's 69,720,375,229,712,477,164,533,808,935,312,303,556,800.)

I don't recall ever being taught this method in school.

Here's one from me, then: prove through induction that the sum from 1 to n of 2n-1 is equal to n².

[blade2k5 1,523]

One of the biggest problems we have today is that students in school are dependent on the net and/or calculators. How many of them can do long division with a pencil and paper?

With my son's generation and now, not very many at all. I taught him basic math before he even started school and he was doing long division before he started kindergarten. When he was in the 3rd or 4th grade [back in '90-'91] one of the many supplies we had to buy him was a calculator, which we did not. When his teacher asked us why we wouldn't supply him with a calculator, my wife simply told her that maybe they should teach the kids how to do math without it. You should have seen the look on her face when she heard that. It went white as a sheet as if she'd seen a ghost. That's when I knew the educational system in this country was getting flushed down the toilet. Nowadays, most kids can't even add or subtract in their heads. Pitiful really.

Back in my day as a student [high school when calculators became widely accessible], if you got caught with one in school, you got suspended for cheating.

[Meg 368]

Back in my day as a student [high school when calculators became widely accessible], if you got caught with one in school, you got suspended for cheating.

In high school, I knew of one guy who had a calculator. It cost about $1,900 in today's dollars. He was told to leave it at home.

I agree that we should know how to do these things without tools but, thinking about it, we did have tools back then. A slide rule is a primitive calculator and, in some classes, its use was totally acceptable. But there is a difference between using a tool and being totally dependent on it.

I haven't looked at a trigonometry book lately but do they still come with those tables in the back? or are students expected to use their calculators instead of looking things up?

[hym 358]

Another surprising relationship that many students have problems with is the one that combines all of mathematics in one equation.

e^iπ-1 = 0

Are you completely sure about this one? My TI-89, Wolfram Alpha, and the University of Toronto's math department don't agree with this formula.

With my son's generation and now, not very many at all. I taught him basic math before he even started school and he was doing long division before he started kindergarten. When he was in the 3rd or 4th grade [back in '90-'91] one of the many supplies we had to buy him was a calculator, which we did not. When his teacher asked us why we wouldn't supply him with a calculator, my wife simply told her that maybe they should teach the kids how to do math without it. You should have seen the look on her face when she heard that. It went white as a sheet as if she'd seen a ghost. That's when I knew the educational system in this country was getting flushed down the toilet. Nowadays, most kids can't even add or subtract in their heads. Pitiful really.

Back in my day as a student [high school when calculators became widely accessible], if you got caught with one in school, you got suspended for cheating.

I'm certainly not going to dispute the benefit of being able to do basic math in your head, but I am hard pressed to understand how not integrating calculators into the teaching process is a good thing. I can speak from personal experience when I say that at the university level, when exam time rolls around, they will expect you to bring a sufficiently powerful calculator and expect you to be extremely proficient with it. If you can't do that, and you run out of time to finish the exam and fail it as a result, they won't feel the least bit of sympathy for you. And why should they? There is no reason to expect everyone to do something by hand when a calculator is capable of doing it nearly instantly. The same principle extends to the work world also. If I'm an employer being asked to choose between two prospective employees, I don't care whether you can do this stuff by hand or not. I care whether you're capable of using the tools in use at the company to get the work done. I don't care how good you are at doing something by hand or in your head; if you can't do it the way it needs to be done by the company, you're worthless.

[blade2k5 1,523]

Another surprising relationship that many students have problems with is the one that combines all of mathematics in one equation.

e^iπ-1 = 0

Are you completely sure about this one? My TI-89, Wolfram Alpha, and the University of Toronto's math department don't agree with this formula.

With my son's generation and now, not very many at all. I taught him basic math before he even started school and he was doing long division before he started kindergarten. When he was in the 3rd or 4th grade [back in '90-'91] one of the many supplies we had to buy him was a calculator, which we did not. When his teacher asked us why we wouldn't supply him with a calculator, my wife simply told her that maybe they should teach the kids how to do math without it. You should have seen the look on her face when she heard that. It went white as a sheet as if she'd seen a ghost. That's when I knew the educational system in this country was getting flushed down the toilet. Nowadays, most kids can't even add or subtract in their heads. Pitiful really.

Back in my day as a student [high school when calculators became widely accessible], if you got caught with one in school, you got suspended for cheating.

I'm certainly not going to dispute the benefit of being able to do basic math in your head, but I am hard pressed to understand how not integrating calculators into the teaching process is a good thing. I can speak from personal experience when I say that at the university level, when exam time rolls around, they will expect you to bring a sufficiently powerful calculator and expect you to be extremely proficient with it. If you can't do that, and you run out of time to finish the exam and fail it as a result, they won't feel the least bit of sympathy for you. And why should they? There is no reason to expect everyone to do something by hand when a calculator is capable of doing it nearly instantly. The same principle extends to the work world also. If I'm an employer being asked to choose between two prospective employees, I don't care whether you can do this stuff by hand or not. I care whether you're capable of using the tools in use at the company to get the work done. I don't care how good you are at doing something by hand or in your head; if you can't do it the way it needs to be done by the company, you're worthless.

You misunderstood what I meant. Grade schoolers don't need calculators, they need to learn how to do math by working the problems out on paper. Jr High or High school I can understand where things get a little tougher, but grade school? Uh-uh, not necessary.

[A Nonny Moose 8,261]

Another surprising relationship that many students have problems with is the one that combines all of mathematics in one equation.

e^iπ+1 = 0

(as amended. ANM)

Are you completely sure about this one? My TI-89, Wolfram Alpha, and the University of Toronto's math department don't agree with this formula.

Well, the old memory is getting a little faint, I guess. I checked and found it is really

e^ix = cos (x) + isin (x)

But try setting x = π and see what you get. You'll need a pretty good calculator to resolve this. I may have got the sign of 1 wrong. It's been a long time.

This relationship stuck in my mind because it contains all the basic mathematical constants. If you want to look it up it is called Euler's Formula.

And yes, I just evaluated that and the -1 belongs on the other side of the = sign. Haven't thought about this in over 50 years and I admit I should have checked. Now you guys have got something to add to your memory kit. I amended the copy. Always good to question things.

  Edited April 19, 2011 by A Nonny Moose  

[Duke87 1,070]

One of the biggest problems we have today is that students in school are dependent on the net and/or calculators. How many of them can do long division with a pencil and paper?

I was taught how to do that in third grade. I still know how to do it, but many others my age have no doubt forgotten, since with the proliferation of calculators, it is a skill that has no practical use anymore.

Do any schools below the post-secondary level teach Euclidean Geometry any more? It is a great discipline and teaches logic and self-reliance.

Yup. 10th grade.

I haven't looked at a trigonometry book lately but do they still come with those tables in the back? or are students expected to use their calculators instead of looking things up?

I definitely had a couple textbooks in middle school that had trigonometric tables in the back. But they were old. A new textbook would not have one because, yes, in the 21st century you "look it up" by punching it into your calculator.

[hym 358]

You misunderstood what I meant. Grade schoolers don't need calculators, they need to learn how to do math by working the problems out on paper. Jr High or High school I can understand where things get a little tougher, but grade school? Uh-uh, not necessary.

It's not about whether the problems are hard enough that a calculator is "necessary." I've gone all the way up to Calculus III and Ordinary Differential Equations and calculators have only infrequently been "necessary." The point is that calculators are an important tool, and children need to be aware of how calculators are capable of helping them (and how to use them properly).

If you wait to introduce calculators till they become "necessary," you've already disadvantaged the child. There is no reason why calculators can't be introduced early, and children can't be expected to be proficient solving the problem "manually" or with a calculator.

Well, the old memory is getting a little faint, I guess. I checked and found it is really

e^ix = cos (x) + isin (x)

That is the equation's alternate form, and one I am well familiar with from my advanced math classes.

But try setting x = π and see what you get. You'll need a pretty good calculator to resolve this.

It's not that hard. My basic Casio fx-115ES got as close as you could expect for not having CAS software installed, and it did it with minimal wait time. Thanks to the CAS software, my TI-89 did it instantly, and got it correct. You can do it by inspection very easily if you remember sine and cosine values for major points on the unit circle. (The cos(x) part is equal to -1, and the sin(x) part is equal to 0, thus negating the i.)

[A Nonny Moose 8,261]

Yup. I misremembered it for its elegance. The degenerate form of the function with pi as the argument is one of the most succinct statements in math. Too bad that most people don't encounter Euler, but you really have to be into the complex number domain before you run into him. A thinker, first class.

Getting back to calculators in school. I carried a slide rule in secondary school. It was most handy in doing reports in Chemistry and Physics classes where you were concentrated on the results and wanted to get the calculations done quickly. However, do remember that a slide rule is a multiplication machine. It doesn't add or subtract. Primary school taught me to do mental arithmetic, and learning about reducing fractions and do algebraic operations on factored expressions made it possible for me do so calculations in my head and devise some quick formulas.

For example: To convert Celsius temperatures to Fahrenheit you simply double the value, subtract 10% and add 32. This is completely accurate. F = (2 x C) - 0.1(2 x C) + 32 is equivalent to (9C)/5 + 32, but doing the fraction in your head is a nuisance, so just factor it.

Tricks like this are not taught in school any more, but in a country like ours, which is equivocal on measurement systems, you have to have some defense. My head is stuffed with all kinds of conversion factors. For example, a kilometer is 5 furlongs (a mile is 8, of course) and there is about three and a third feet in a meter. This is pretty good too, when you find that 15m is very close to 50ft.

[blade2k5 1,523]

You misunderstood what I meant. Grade schoolers don't need calculators, they need to learn how to do math by working the problems out on paper. Jr High or High school I can understand where things get a little tougher, but grade school? Uh-uh, not necessary.

It's not about whether the problems are hard enough that a calculator is "necessary." I've gone all the way up to Calculus III and Ordinary Differential Equations and calculators have only infrequently been "necessary." The point is that calculators are an important tool, and children need to be aware of how calculators are capable of helping them (and how to use them properly).

If you wait to introduce calculators till they become "necessary," you've already disadvantaged the child. There is no reason why calculators can't be introduced early, and children can't be expected to be proficient solving the problem "manually" or with a calculator.

You're still not getting my point hym. The point is, kids need to learn how to do the problems first using only their brains before you start throwing electronical devices at them to do all the solving for them. It's no wonder why most high school grads in this country can't do simple adding/subtracting in their heads or make change from a dollar without a calculator or cash register. You do realize how pathetic that is right?

[hym 358]

You misunderstood what I meant. Grade schoolers don't need calculators, they need to learn how to do math by working the problems out on paper. Jr High or High school I can understand where things get a little tougher, but grade school? Uh-uh, not necessary.

It's not about whether the problems are hard enough that a calculator is "necessary." I've gone all the way up to Calculus III and Ordinary Differential Equations and calculators have only infrequently been "necessary." The point is that calculators are an important tool, and children need to be aware of how calculators are capable of helping them (and how to use them properly).

If you wait to introduce calculators till they become "necessary," you've already disadvantaged the child. There is no reason why calculators can't be introduced early, and children can't be expected to be proficient solving the problem "manually" or with a calculator.

You're still not getting my point hym. The point is, kids need to learn how to do the problems first using only their brains before you start throwing electronical devices at them to do all the solving for them. It's no wonder why most high school grads in this country can't do simple adding/subtracting in their heads or make change from a dollar without a calculator or cash register. You do realize how pathetic that is right?

I understand what your point is, but I do not agree with it. It is my opinion that electronic tools need to be introduced as a part of the learning process. Teach them how to solve the problem, then teach them how it can be solved using a calculator or whatever other tool is being used. There is no reason that we can't expect children to demonstrate proficiency doing it manually or with an electronic aid.

Assuming I have children (and my wife isn't against the idea), I full well plan to teach my children basic math, then hand them a calculator, and expect them to learn how to do it that way. By the time a junior high student was learning how to use a calculator, my children would be more than familiar with how it can be used on all their math and science homework.

[Meg 368]

There is no reason that we can't expect children to demonstrate proficiency doing it manually or with an electronic aid.

If you say "and" instead of "or", I agree with you.

Blade has a point: without being able to do basic math in their heads, people can not make change as a cashier. Solid coins and paper money haven't disappeared yet.

[hym 358]

There is no reason that we can't expect children to demonstrate proficiency doing it manually or with an electronic aid.

If you say "and" instead of "or", I agree with you.

Blade has a point: without being able to do basic math in their heads, people can not make change as a cashier. Solid coins and paper money haven't disappeared yet.

Blame the vision in my head for the choice of "or" instead of the more clear "and".

Point is, I should be able to hand a kid a calculator, say "work these problems," and he should be able to solve them with no trouble. I should also be able to take the calculator away, hand him a new set of problems, say "do these without the calculator," and he should be able to solve them quickly and accurately.

[Duke87 1,070]

On the other hand, unless you're working in a small shop that has a very old or very basic cash register, the register itself is capable of calculating change for you.

Although I agree it's pathetic if you can't make change in your head. We're talking about subtracting from 100 here! Not complicated stuff!

[Meg 368]

Blame the vision in my head for the choice of "or" instead of the more clear "and".

Point is, I should be able to hand a kid a calculator, say "work these problems," and he should be able to solve them with no trouble. I should also be able to take the calculator away, hand him a new set of problems, say "do these without the calculator," and he should be able to solve them quickly and accurately.

That works.

On the other hand, unless you're working in a small shop that has a very old or very basic cash register, the register itself is capable of calculating change for you.

Although I agree it's pathetic if you can't make change in your head. We're talking about subtracting from 100 here! Not complicated stuff!

I have seen a register compute that I needed 40 cents in change and watched the teenager look at the cash drawer and blink. They have to stop and figure out which combination of coins will result in that amount.

I have also, for instance, owed $10.15 and handed them a $20 and a quarter. Some of them do not know how to handle that.

It is pathetic.

[astronelson 339]

[quote name='Meg' timestamp='1303354936' post='1147952'

I have seen a register compute that I needed 40 cents in change and watched the teenager look at the cash drawer and blink. They have to stop and figure

[blade2k5 1,523]

I have seen a register compute that I needed 40 cents in change and watched the teenager look at the cash drawer and blink. They have to stop and figure out which combination of coins will result in that amount.

I have also, for instance, owed $10.15 and handed them a $20 and a quarter. Some of them do not know how to handle that.

It is pathetic.

I'd say about 60-70% of students here where I live grow into adulthood not knowing how to make change without a register telling them what change to give back. The percentage goes even higher with the example you gave, especially if it's say $10.18 and you give them a $20 and 3 pennies. Throws them for a loop every time. God forbid if the power goes out and they can't use the register.

[A Nonny Moose 8,261]

Poor darlings. Just imagine how they would be fixed if the net went off the air for an hour. They wouldn't be able to tweet, and most of them don't know how to whistle a happy tune. It would be kind of like the wooden whistle that wouldn't whistle.

I wonder if there is going to be a psychological diagnosis on lack of tweeting? Maybe it would be called the Sylvester syndrome.

I t'ought I taw a puddy tat.

[Duke87 1,070]

I have seen a register compute that I needed 40 cents in change and watched the teenager look at the cash drawer and blink. They have to stop and figure out which combination of coins will result in that amount.

Okay, ouch. A round number even!

I have also, for instance, owed $10.15 and handed them a $20 and a quarter. Some of them do not know how to handle that.

I would blame the fact that it is unusual for people to pay for things in this fashion. Usually people paying in cash pay with just bills, and then go to the CoinStar machine every so often to turn all the coins they get as change into bills. I refuse to pay a fee just to have cash changed, so I don't do this, but keeping change from building up can be difficult. You need to find excuses to use it and remember to use it. I am not normally in the habit of carrying coins in my pocket, but I do have a sizable stash of them in the tray in my car, which I can then take with me if I remember to in order to get rid of some of it. Of course, I wouldn't put a cashier in the situation you describe. When I bring change into the store with me, I count out three quarters, two dimes, a nickel, and four pennies. Why this combination? Well, with those coins, you can assemble any cent amount from .00 to .99, which means I can pay exactly and get no change, or at least pay the cents part accurately and get only bills in change. Remember, I'm looking to get rid of coins, here!

On a related note, this comic.

Win. That's an engineer's logic alright.

For anyone who doesn't get it:

$7.14 - $1.89 = $5.25, change which can be given with just one bill and one coin. If Dilbert were the cashier, he'd consider that making his job easy.

The percentage goes even higher with the example you gave, especially if it's say $10.18 and you give them a $20 and 3 pennies. Throws them for a loop every time.

$20.03 - $10.18 = $9.85. You're just trying to make the cashier's life difficult, aren't you?

God forbid if the power goes out and they can't use the register.

The least of their worries. Power outages also render inoperable things like lights, refrigerators, elevators and escalators, barcode scanners, etc... to the point where stores are almost universally forced to close if the power goes out.