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What is the sum of all the digits in all the integers from one to one million?

The Power of the Sum

Age 14 to 16 ShortChallenge Level

Answer: $n=7$

Using pairs of equal numbers
\begin{align}&2^6+2^5+2^4+2^4\\ =&2^6 + 2^5 + 2^4\times2\\ =&2^6 + 2^5 + 2^{4+1}\\ =&2^6 + 2^5 + 2^5\\ =&2^6 + 2^5\times2\\ =&2^6 + 2^6\\ =&2^7\end{align}

Finding the value of the powers of 2
$2^2=4$                 $2^5=32$
$2^3=8$                 $2^6=64$
$2^4=16$               $2^7=128$

So $2^6+2^5+2^4+2^4=64+32+16+16=128=2^7$

Factorising and using index laws
Notice that all of the numbers in the sum are multiples of $2^4$, since $2^6=2^2\times2^4,2^5=2\times2^4,2^4=1\times2^4.$ So
\begin{align}2^6+2^5+2^4+2^4&=2^2\times2^4+2\times2^4+1\times2^4+1\times2^4\\ &=\left(2^2+2+1+1\right)\times2^4\\ &=\left(4+2+1+1\right)\times2^4\\ &=8\times2^4\\ &=2^3\times2^4\\ &=2^7\end{align}
You can find more short problems, arranged by curriculum topic, in our short problems collection.