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14 Divisors

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Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Weekly Problem 16 - 2010

Stage: 2 and 3 Short Challenge Level: Challenge Level:1
If the first two digits are $a$ and $b$ (with $a\neq 0$) then the six terms will be $a$, $b$, $a+b$, $a+2b$, $2a+3b$ and $3a+5b$, so we must have $3a+5b< 9$.

If $b=0$ then $a$ can be $1$, $2$ or $3$. If $b=1$ then $a$ can only be $1$, and $b$ cannot be greater than $2$.

Hence there are four possibilities, namely $101123$, $202246$, $303369$ and $112358$.

This problem is taken from the UKMT Mathematical Challenges.

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