2017 AIME I Problem 2

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