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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?

Factor List

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

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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