Doubly consecutive sums
How many numbers less than 2017 are both the sum of two consecutive integers and the sum of five consecutive integers?
Age
11 to 14
Challenge level
Problem
How many numbers less than $2017$ are both the sum of two consecutive positive integers, and the sum of five consecutive positive integers?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: $201$
Sum of 2 consecutive numbers is $n + (n+1)=2n+1$
Sum of 5 consecutive numbers is $m + (m+1)+(m+2)+(m+3)+(m+4) = 5m+10$
($m$ and $n$ different numbers because the sums have different starting points)
$2n+1=5m+10$
Can find $n$ from any $m$, as long as $5m+10$ is odd
$5m+10$ is odd whenever $m$ is odd
And $5m+10 \lt 2017$
$\therefore 5m \lt 2007\\
\therefore m \le 401$
There are $201$ odd numbers up to $401$