The least number which when divided by 4, 7, 9, 11, and 13 leaves the same remainder 1 in each case, is:

We want the least number N such that: N leaves remainder 1 when divided by 4, 7, 9, 11, and 13. That means: N ≡ 1 (mod 4) N ≡ 1 (mod 7) N ≡ 1 (mod 9) N ≡ 1 (mod 11) N ≡ 1 (mod 13) So N − 1 is divisible by all of these numbers. Therefore: N − 1 = LCM(4, 7, 9, 11, 13) Now find the LCM: 4 = 2² 7 = 7 9 = 3² 11 = 11 13 = 13 LCM = 2² × 3² × 7 × 11 × 13 Compute step by step: 2² = 4 3² = 9 4 × 9 = 36 36 × 7 = 252 252 × 11 = 2772 2772 × 13 = 36036 So: N − 1 = 36036 N = 36036 + 1 = 36037 Final answer: N = 36037