Show that 2^ 4n+4 -15n-16, where n is divisible by 225. - Vidyadip

Question

Show that $2^{4\text{n}+4}-15\text{n}-16,$ where $\text{n}\in\text{N}$ is divisible by 225.

✓

Answer

$2^{4\text{n}+4}- 15\text{n}-16=2^{4(\text{n}+1)}-15\text{n}-15-1$
$=(16)^{\text{n}+1}-15(\text{n}+1)-1$
$=(1+15)^{\text{n}+1}-15(\text{n}+1)-1$
$=\big[{^{\text{n}+1}\text{C}}_0+{^{\text{n}+1}\text{C}}_1(15)+{^{\text{n}+1}\text{C}}_2(15)^2+.....\\+{^{\text{n}+1}\text{C}}_{\text{n}+1}(15)^{\text{n}+1}\big]-15(\text{n}+1)-1$
$=\big[1+15({\text{n}+1})+{^{\text{n}+1}\text{C}}_2(15)^2+.....\\+{^{\text{n}+1}\text{C}}_{\text{n}+1}(15)^{\text{n}+1}\big]-15(\text{n}+1)-1$
$=225\big[{^{\text{n}+1}\text{C}}_2+.....{^{\text{n}+1}\text{C}}_{\text{n}+1}(15)^{\text{n}-1}\big]$
= 225 × natural number

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