CAT 1998 — QA Question 17
LCM & HCFEasy
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?
Answer & solution
Correct answer: 1
- A
0
1
- C
2
- D
None of these
Solution
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