Remainder, you are awesome (2) ....

Today I happened to have an encounter with the same type of question that I had had in my blog. The present blog is the extension of previous blog on the same issue.

The question is:

A is the set of positive integers such that when divided by 2, 3, 4, 5, 6 leaves the remainder 1, 2, 3, 4, 5 respectively. How many integers between 0-100 belong to set?

A. 0 B. 1 C. 2 D. 3

I solved it like this:

Let the number be P. Then, as given:

P = 1 mod 2, i.e. P = -1 mod 2

P = 2 mod 3, i.e. P = -1 mod 3

P = 3 mod 4, i.e. P = -1 mod 4

P = 4 mod 5, i.e. P = -1 mod 5

P = 5 mod 6, i.e. P = -1 mod 6

Means, the number leaves a remainder -1 in each case when divided by 2, 3, 4, 5, 6 .

Hence, the number is: LCM (2, 3, 4, 5, 6) + (-1) = LCM {LCM (2,3,4), LCM (5,6)} - 1

= LCM {12, 30} - 1 = 60 -1 = 59

The required number is: 59. Option B is the answer.

Remainders are really awesome.

{Written on www.mathsacad99.com by Sanjay Mohan Bhatnagar}