Question

When 3444 and 4454 are divided by a three digit number. The same remainder is left. The smallest three digit number is?

Answer 1

Let the smallest 3 digit number which divides 3444 and 4454 be n, and which leaves equal remainder r in each case. Hence;

4454 == r mod n; and 3444 == r mod n

Subtracting second from first, we have, 1010 == 0 mod n

Thus, when the given numbers are divided by n, they leave same remainder equal to 0. Hence every factor of 1010 divides the given numbers completely too.

But we need a smallest 3 digit number, hence, 101, which is factor of 1010, is the smallest 3 digit number which divides 3444 and 4454 and leaves equal remainder in each case. The remainder is: 10, as, 3444 == 10 mod 101, and 4454 == 10 mod 101.