Question

When each of $702,$ $787,$ and $855$ is divided by the positive integer $m,$ the remainder is always the positive integer $r.$ When each of $412,$ $722,$ and $815$ is divided by the positive integer $n,$ the remainder is always the positive integer $s \neq r.$ Find $m + n + r + s.$

Answer

Answer: 62

Concepts:greatest common divisor modular arithmetic divisibility

Difficulty rating: 2070

Small Hint:

If two numbers leave the same remainder when divided by $m,$ then $m$ divides their difference

Big Hint:

So $m$ is a common factor of $85$ and $68,$ and $n$ is a common factor of $310$ and $93$

Solution:

Numbers leaving equal remainders upon division by $m$ differ by multiples of $m,$ so $m$ divides both $787 - 702 = 85$ and $855 - 787 = 68.$ Since $\gcd(85, 68) = 17$ and $m$ must exceed the positive remainder $r,$ we get $m = 17,$ and $r = 702 - 41 \cdot 17 = 5.$

Similarly $n$ divides both $722 - 412 = 310$ and $815 - 722 = 93,$ and $\gcd(310, 93) = 31,$ so $n = 31$ and $s = 412 - 13 \cdot 31 = 9,$ which indeed differs from $r.$

The requested sum is $17 + 31 + 5 + 9 = 62.$

2017 AIME I Problem 2

Answer: 62

Difficulty rating: 2070

Small Hint:

Big Hint:

Solution:

Problem 2 in Other Years