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14 Divisors

What is the smallest number with exactly 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?

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Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Weekly Problem 46 - 2010

Stage: 3 Short Challenge Level: Challenge Level:1

The didgits $1$ and $3$ will always be followed by the digit $2$. The digit $2$ can be followed by either $1$ or $3$. Hence the digit $2$ appears exactly five times in a ten digit number, in alternate positions.
If the first digit is $2$, then in each even position we have two choices, $1$ or $3$. This gives $2\times 2\times 2\times 2\times 2 = 32$ possibilities. Otherwise, the second digit is $2$ and in each odd position we have two choices. So again there are $32$ possibilities, making a total of $64$

This problem is taken from the UKMT Mathematical Challenges.

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