Find the smallest number which leaves 22, 35, 48 and 61 as remainders when…

Key insight: each given remainder is 4 less than its divisor, so adding 4 to the unknown number makes it divisible by all the divisors.

Therefore n + 4 is a common multiple of 26, 39, 52 and 65. Find the least common multiple (LCM) of these numbers.

• 26 = 2 × 13

• 39 = 3 × 13

• 52 = 2^2 × 13

• 65 = 5 × 13

Take the highest powers of the prime factors: 2^2, 3, 5 and 13. So LCM = 2^2 × 3 × 5 × 13 = 4 × 3 × 5 × 13 = 780.

Thus n + 4 = 780, which gives n = 780 - 4 = 776. This is the smallest positive number satisfying the given remainders.