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Power Crazy

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?



If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation? If in addition you know that the lines $AB$ and $CD$ are parallel which angles can you find?


Congratulations on your solutions to Ian Walker, Owen Jones and Tom Embury (Y7) St James Middle School, Bury St Edmunds, to students from Y9 and Y10, The Mount School, York and to Shabbir Tejani, age 13, Jack Hunt School, Peterborough. Here is Shabbir's solution:

Angle $G = 180$ - (angle $Y$ + angle $X$)
Angle $E = 180$ - angle $G$
Angle $F = 180$ - (angle $E$ + angle $Z$)
Angle $J =$ angle $F$
Angle $H = 180$ - angle $F$
Angle $I = 360$ - (angle $X$ + angle $H$ + angle $G$)
Angle $O = 180$ - angle $I$
Angle $N = 180$ - angle $O$

These are all the angles that can be found without additional information. If we are given the added information that the lines $AB$ and $CD$ are parallel then we know:

Angle $M =$ angle $Z$
Angle $L = 180$ - (angle $M$ + angle $N$)
Angle $P = 180$ - angle $L$
Angle $K = 180$ - angle $G$
Angle $Q = 180$ - angle $P$
Angle $R = 180$ - angle $K$
Angle $T = 180$ - angle $R$

Angle S and U cannot be found.

HOWEVER if lines $CC'$ and $AD$ are parallel, then angle $M =$ angle $S$. Therefore Angle $U = 180$ - (angle $S$ + angle $T$).