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The Power of the Sum

Age 14 to 16 Short
Challenge Level

Answer: $n=7$

Using pairs of equal numbers
=&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\\

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
You can find more short problems, arranged by curriculum topic, in our short problems collection.