Five-to
The number 2005 is the sum of a sequence of five consecutive positive integers. What is the smallest of these integers?
Age
11 to 14
Challenge level
Problem
The number 2005 is the sum of a sequence of five consecutive positive integers.
What is the smallest of these integers?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: $399$
Using algebra
Let the five consecutive positive integers be $x-2$, $x-1$, $x$, $x + 1$, $x + 2$. Their sum is $5x$, so $5x = 2005$, that is $x = 401$. The five numbers are therefore $399, 400, 401, 402, 403,$ so the smallest number is $399$.
Using approximation/averages
The numbers are close to $2005\div5 = 401$
$$401+401+401+401+401=2005\\
401+400+401+402+401=2005\\
399+400+401+402+403=2005$$ The smallest number is $399$