# Solution: Finding A and B using Number Theory

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**Published on Aug 9, 2018**- Let's use our knowledge of the sum of factors of an integer to simplify the problem and examine the resulting expression by casework.

Congratulations to Gabriel N., Allaizn, Benjamin Wang, Essentials of Math, iQuickdraw X, staffehn, Quwertyn, NoName, Hung Hin Sun, and Jacob Glidewell for successfully solving this math challenge question! Gabriel N. was the first person to solve the question.

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MountainC6 days agoShame. I was almost gonna try a quadruple summation lol

Abdou Kabar2 months agoðŸ˜ðŸ˜ðŸ˜ðŸ˜

Pavel Vanak10 months ago^{+3}Why 42? because it's the answer to everything in the universe. Knew it!

Suaqe10 months ago^{+1}Am I a nerd now

Alberto Zordan10 months ago^{+4}There's one subtle thing that's eluding me: the restriction on the exponents x,y,w,z. I know that, for those numbers to be divisible by 6, they have to be >=1, but doing so one can skip some factors (e.g. 90 is divisible by 6 but has also 2 and 3 as factors). However, when you write down the sum of all possible factors of A and B, you also rightly include 1, which is 2^0 or 3^0. So here the exponents are

LetsSolveMathProblems10 months ago^{+3}The exponents of 2 and 3 in the prime factorization of A and B must be greater than or equal to 1. The exponents of 2 and 3 in the prime factorization of a FACTOR of A or B can be anything (even 0) as long as it divides A or B.

GreenMeansGO10 months ago^{+3}Here is how I did it.

After trying (6,12) I realized that the sum of a*b values equals

(1+2+3+6)(1+2+3+6+12) = Ïƒ(6)*Ïƒ(12) = 12*28 = 336.

I used Ïƒ values and found (6,72) since

Ïƒ(6)*Ïƒ(72)=12*195=2340

But checking for numbers in between, I found (18,24)

Ïƒ(18)*Ïƒ(24)=39*60=2340.

No other pair works. So the answer is 18+24=42.

P.S. A and B must be distinct since 2340 is not a perfect square. Otherwise

2340 = Ïƒ(A)^2.

Mike Onega10 months ago^{+2}where can I go to learn this I feel dumb that I don't know this?

carlos takeshi10 months agoAOPS books are really helpful, u can also practice w AIME/late AMC 12 questions

Lova aaa10 months agoJust buy a good book and practice a bit

GreenMeansGO10 months ago^{+2}I learned this stuff from college classes, especially Number Theory.

Oliver Hees10 months ago^{+5}6:58 nice editing