(a) The numbers 11284 and 7655, when divided by a certain 3-digit number, leave the same remainder. Find the number and the remainder. (b) What is the least number to be subtracted from 1936 so that when divided by 9, 10 and 15 it will leave the same remainder in each case?

For part (a), since both numbers leave the same remainder when divided by the unknown 3-digit number, their difference $$11284-7655=3629$$ must be exactly divisible by it; factoring $$3629=19\times 191$$ shows 191 is the only 3-digit factor, and dividing 7655 by 191 gives a remainder of 15. For part (b), the LCM of 9, 10 and 15 is 90, and dividing 1936 by 90 gives quotient 21 with remainder 46; since the largest multiple of 90 below 1936 is $$90\times 21=1890$$, adding the maximum possible common remainder of 8 to this gives the target number 1898, so the least number to subtract is $$1936-1898=38$$.