Marcab wrote:
Consider a sequence of numbers given by the expression 5 + (n - 1) * 3, where n runs from 1 to 85. How many of these numbers are divisible by 7?
A. 5
B. 7
C. 8
D. 11
E. 12
It’s relatively fast and easy to use modular arithmetic in solving this question
The given arithmetic sequence should be divisible by 7 and we need to find the values of \(n\) which will satisfy this.
So we have:
5 + (n-1)*3 ≡ 0 (mod 7)
(n-1)*3 ≡ -5 (mod 7)
(n-1)*3 ≡ 9 (mod 7)
n-1 ≡ 3 (mod 7)
n ≡ 4 (mod 7)
Our \(n\) leaves remainder 4 when divided by 7 and has following form:
\(n = 7*x+4\)
This is arithmetic progression which starts with 4 and ends with 81 with common difference 7.
Total number of elements in it is:
\(\frac{(81-4)}{7}+1=12\)
Answer E.