### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### 14 Divisors

What is the smallest number with exactly 14 divisors?

### 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?

# Factor List

##### Stage: 3 Short Challenge Level:

Let the smallest prime factor of N be p, whence the second largest factor, and the highest factor tina wrote down, is ${N}\over{p}$.

Now we have $\frac{N}{p}= 45p$. So ${N}= 45 p^2$.

Since N is a multiple of $45$, it has prime factors of $3$ and $5$ and, because p is the smallest prime factor of N, we can conclude that p can be only $2$ or $3$.

Hence either $N = 45\times 2^2 = 180$ or $N = 45\times 3^2 = 405$

This problem is taken from the UKMT Mathematical Challenges.
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