Answer :

Solution :

${\displaystyle \frac{{11}^{52}-1}{{11}^{13}+1}}={\displaystyle \frac{({{11}^{13})}^4-1^4}{{11}^{13}+1}}$

⇒${\displaystyle \frac{{11}^{52}-1}{{11}^{13}+1}}={\displaystyle \frac{[{({11}^{13})}^2-1^2][{({11}^{13})}^2+1^2]}{{11}^{13}+1}}$

⇒${\displaystyle \frac{{11}^{52}-1}{{11}^{13}+1}}={\displaystyle \frac{[{11}^{13}-1][{11}^{13}+1][{({11}^{13})}^2+1^2]}{{11}^{13}+1}}$

⇒${\displaystyle \frac{{11}^{52}-1}{{11}^{13}+1}}=[{11}^{13}-1][({{11}^{13})}^2+1]$

∴ (1152 - 1) is exactly divisible by (1113 + 1)