Find the number of integers N from 1 to 100 such that \(N^2 + 8N + 15 \) is a multiple of 7.

Derekshackk Jun 23, 2022

#1**+2 **

We can factor N^2+8N+15 into (N+3)(N+5). So if we want N^2+8N+15 to be divisible by 7, we need either n+3 or N+5 divisible by 7. The list of N such that N+3 is divisible by 7 is: 4,11,18,25,...,95, which is (95-4)/7+1=14 numbers. The list of N such that N=5 is divisible by 7 is 2, 9, 16, 23,...,100 which is (100-2)/7+1=15 numbers. 14+15=\(\boxed{29}\)

SparklingWater2 Jun 23, 2022

#1**+2 **

Best Answer

We can factor N^2+8N+15 into (N+3)(N+5). So if we want N^2+8N+15 to be divisible by 7, we need either n+3 or N+5 divisible by 7. The list of N such that N+3 is divisible by 7 is: 4,11,18,25,...,95, which is (95-4)/7+1=14 numbers. The list of N such that N=5 is divisible by 7 is 2, 9, 16, 23,...,100 which is (100-2)/7+1=15 numbers. 14+15=\(\boxed{29}\)

SparklingWater2 Jun 23, 2022