There is such a positive integer: if it is divided by 5, the remainder is 3; if it is divided by 6, the remainder is 4; if it is divided by 7, the remainder is 1. Find the least possible value of such number.
Question
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Answer:
- The least possible number is 148
—————————————————Let the number be N.If N is divided by 5 the remainder is 3:- N = 5k + 3,
Similarly:- N = 6p + 4,
- N = 7q + 1.
We can observe N + 2 is divisible by both 5 and 6, then it can be represented as:- N + 2 = 30m,
so- 30m = 7q + 1 + 2 or
- 30m – 3 = 7q
The left side is divisible by 3, then q = 3t:- 30m – 3 = 7*3t
- 10m – 1 = 7t
The least m and t would be m = 5 and t = 7.Then:- N + 2 = 30*5 = 150 ⇒ N = 148 is the least possible number.
Proof:- 148/5 = 29 (rem 3)
- 148/6 = 24 (rem 4)
- 148/7 = 21 (rem 1)