A is the set of positive integers such that when divided by 2, 3, 4, 5, 6 leaves the remainders 1, 2, 3, 4, 5 respectively. How many integers between 0 and 100 belong to set A?
Explanation:
Note that the difference between the divisors and the remainders is constant. 2 – 1 = 3 – 2 = 4 – 3 = 5 – 4 = 6 – 5 = 1 In such a case, the required number will always be [a multiple of LCM of (2, 3, 4, 5, 6) – (The constant difference)]. LCM of (2, 3, 4, 5, 6) = 60 Hence, the required number will be 60n – 1. Thus, we can see that the smallest such number is (60 × 1) – 1 = 59 The second smallest is (60 × 2) – 1 = 119 So between 1 and 100, there is only one such number, viz. 59
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