What least number must be subtracted from 2963 so that the resulting number when divided by 9, 10 and 15, the remainder in each case is 5?

We need a number N such that when (2963 - N) is divided by 9, 10, and 15, the remainder is always 5. LCM of 9, 10, 15 = 90. If a number leaves remainder 5 when divided by 90, it will also leave remainder 5 when divided by any factor of 90. So we need: (2963 - N) ≡ 5 (mod 90), meaning 2963 - N = 90k + 5 for some k. 2963 ÷ 90 = 32 remainder 83. So 2963 = 90×32 + 83. We want this to be 90k + 5, so we need to subtract: 83 - 5 = 78. Therefore N = 78.