The last digit of 220062^{2006}

My 13 year old son was asked this question in a maths challenge. He correctly guessed 4 on the assumption that the answer was likely to be the last digit of 26. However is there a better explanation I can give him?

Answer

2^{4} = 16. Multiply any even integer by 6 and you don’t change the last digit:
0 \times 6 = 0, 2 \times 6 = 12, 4 \times 6 = 24 etc. The same is true if you multiply an even integer by anything whose last digit ends in 6, in particular by 16. Now
2006 = 2004 + 2 where 2004 = 501 \times 4, so
2^{2006} = (2^4)^{501} \times 2^2 has the same last digit as 2^2.

Attribution
Source : Link , Question Author : Keith Miller , Answer Author : Robert Israel

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