Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11, 15 respectively.

According to the condition given in the question

On dividing 398 by the required number, there is a remainder of 7.

This means that 398 – 7 = 391, is exactly divisible by the required number.

Similarly, 436 – 11 = 425 and 542 – 15 = 527 are exactly divisible by the required number.

The H.C.F of two positive integers is the largest positive integer that divides both the integers.

So, the required number will be the H.C.F of 391, 425 and 527.

And that can be found by using Euclid’s division algorithm.

425 = 391 x 1 + 34

391 = 34 x 11 + 17

34 = 17 x 2 + 0

Thus, H.C.F = 17

Hence, the required number is 17.