### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

# Factor List

##### Stage: 4 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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