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January 24

Greatest number which cannot be expressed as a sum of 3m + 20n, where m and n are whole numbers?

As title - I can't quite recall the name of this type of problem, which makes it hard to look up information on. Any help appreciated, thanks! --131.111.135.84 (talk) 11:37, 24 January 2008 (UTC)[reply]

Any whole number N can be represented by 3·m+20·n, because 3·7+20·(−1) = 21−20 = 1 and so 3·(7·N)+20·(−N) = 1·N = N. Bo Jacoby (talk) 12:06, 24 January 2008 (UTC).[reply]

If you replace whole numbers for naturals, the same is true again. No such greatest number could exist, because if it existed (let's call it x), x+23 couldn't be expressed as such sum either. Pallida Mors 12:34, 24 January 2008 (UTC)[reply]

Huh? 1 cannot be expressed in this way but 24 can (or, if you don't allow 0, 24 vs. 47). The largest number for which this is impossible is 37, because for any number

x\geq 40

, either

x,x-20

or

x-40

is divisible by 3. -- Meni Rosenfeld (talk) 12:47, 24 January 2008 (UTC)[reply]

You're of course right, Meni. I thought that, if

x+23=3m+2n

Then,

x=3(m-1)+2(n-1)

. Absurd.

But this is not absurd, as (m-1) could then perfectly be 0 (or -1). Pallida Mors 13:23, 24 January 2008 (UTC)[reply]

More generally, for a and b whole numbers, the numbers which can be expressed as am + bn are the numbers divisible by gcd(a,b) (i.e. the numbers {gcd(a,b)k | k in Z}) where gcd is the greatest common divisor. This follows since by the extended Euclidean algorithm there exists whole numbers m and n such that gcd(a,b) = am+bn. In particular, if gcd(a,b) = 1 (which is the case, when a=3 and b=20) all whole numbers can be expressed as am+bn Aenar (talk) 12:58, 24 January 2008 (UTC)[reply]

The case where m and n are restricted to natural numbers is an example of a Frobenius coin problem. Gandalf61 (talk) 13:04, 24 January 2008 (UTC)[reply]

Sorry, I meant m and n were positive whole numbers... so the answer is 37? Thanks. --131.111.8.97 (talk) 15:47, 24 January 2008 (UTC)[reply]

That's the answer if you allow m and n to be zero. If they have to be whole numbers >0, the answer becomes 60. Algebraist 15:53, 24 January 2008 (UTC)[reply]

Is it just me, or does this look more like a homework problem than anything else? Pdbailey (talk) 18:14, 25 January 2008 (UTC)[reply]

I hope not. I can't think of any class for which this would be good homework. Black Carrot (talk) 19:00, 25 January 2008 (UTC)[reply]

It was the second post that looked homeworky to me. But both question posts are from University of Cambridge, it's an awfully sad question for a college class. Pdbailey (talk) 19:41, 25 January 2008 (UTC)[reply]

This question is not a set problem for any Cambridge maths course at present. I can't speak for other subjects. Algebraist 20:15, 25 January 2008 (UTC)[reply]

POUNDS

WHY IS LBS USED AS AN ABBREVIATION FOR POUNDS? —Preceding unsigned comment added by 69.19.14.33 (talk) 21:38, 24 January 2008 (UTC)[reply]

Blame in on the Romans. And please don't post in all-CAPS. --LarryMac | Talk 21:41, 24 January 2008 (UTC)[reply]

The romans would do that because they had a system but I forgot--Kop the man (talk) 03:00, 25 January 2008 (UTC)loperman2510[reply]

LBS is an abbreviation for Pound-Mass. I cannot find the long world for lbs on googles anywhere. Daniel5127 ^(talk) 05:45, 25 January 2008 (UTC)[reply]

No, "lb." is an abbreviation for "pound", and "lbs." is an abbreviation for "pounds", which come from the Latin word "libra", as LarryMac said. It was used as an abbreviation for pound long before people learned to distinguish between weight and mass. In situations where it is important to distinguish between the two, I have seen "lbf" used as an abbreviation for "pound (of) force". —Bkell (talk) 06:48, 25 January 2008 (UTC)[reply]

More precisely, "lb." means "pound" or "pounds", and "lbs." is another abbreviation for "pounds". Both styles are commonly used for the plural. --Anonymous, 02:23 UTC, January 26, 2008.

"Libra" is also the Latin word for scales or balance, hence the name of the constellation and the astrological sign. The connection between a weight and scales should be readily apparent. -- JackofOz (talk) 04:58, 26 January 2008 (UTC)[reply]