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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:
$$f'(x)=1$$