Number 5

Number Theory Level 3

There are two positive integers $X$ and $Y$ . When $X$ is divided by 237, the remainder is 192. When $Y$ is divided by 117, the quotient is the same but the remainder is 108. Find the remainder when the sum of $X$ and $Y$ is divided by 118.

The answer is 64.

2 solutions

Soumya Shrivastva
May 16, 2016

X=237q+192, where q is a positive integer and,
Y=117q+108
then, X+Y=354q=300, so by putting different values of q we will get the values of X+Y.
So, values of X+Y are:300, 654, 1008,......so on
Now, 300 leaves remainder 64 when divided by 118 and this is same for 654, 1008,...........
Hence, any value of X+Y gives remainder 64 when divided by118

After we get x+y = 354q+300... we can easily see that 354 is exact multiple of 118.. instead of substituting values we can easily ignore that term while taking remainder.. just consider 300 % 118 = 64😉

karan shah - 5 years ago

Exactly! Did the same

Aditya Kumar - 5 years ago

E Koh
Nov 3, 2019

Let the quotient obtained in both the cases be n. X = 237 n + 192 Y = 117 n + 108 Adding both the equations, X + Y = 354 n + 300 = 118 ( 3 n ) + 118 ( 2 ) + 64 =118 ( 3 n + 2 ) + 64. Therefore, the remainder obtained when sum of X & Y is divided by 118 is 64.

Number 5

The answer is 64.

2 solutions

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