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## Monday, 16 June 2014

### Sunday Afternoon Maths XVII Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

#### Maths Jam

15th February 2022.
##### Extension
What is the latest date in the month on which Maths Jam can fall and when will this next happen?

#### N

(b) implies that the digits of $$N$$ are all 1 or 7, so $$N$$ can only be 111, 117, 171, 177, 711, 717, 771 or 777. These are all divisible by 3, so no such integers $$N$$ exist.
##### Extension
Consider 21-digit integers $$N$$ such that:
(a) $$N$$ is not exactly divisible by 2, 3 or 5.
(b) No digit of $$N$$ is exactly divisible by 2, 3 or 5.
How many such integers $$N$$ are there?

#### Square Numbers

Let $$a^2$$ and $$b^2$$ be the two square numbers.
$$2(a^2 +b^2 ) = 2a^2 +2b^2$$ $$= a^2 + 2ab + b^2 + a^2 - 2ab + b^2$$ $$= (a+b)^2 +(a-b)^2$$
##### Extension
Prove that 3 times the sum of 3 squares is also the sum of 4 squares.

#### Differentiate This

$$f(x)=e^{x^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}} }$$ $$=e^{e^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}\ln{x}} }$$ $$=e^{e^{ \ln{\left(\ln{x}\right)}} }$$ $$=e^{\ln{x} }$$ $$=x$$
Therefore:
$$f'(x)=1$$