Question

Find the total number of divisors, sum of all factors and product of all factors of $11760$.

• A. Total number of factors : $40$
  Sum of factors : $42440$
  Product of factors : ${11760}^{20}$
• B. Total number of factors : $60$
  Sum of factors : $42480$
  Product of factors : ${11760}^{20}$
• C. Total number of factors : $40$
  Sum of factors : $42440$
  Product of factors : ${11760}^{20}$
• D. Total number of factors : $60$
  Sum of factors : $42408$
  Product of factors : ${11760}^{40}$

Answer 1

The correct option is D Total number of factors : $60$

Sum of factors : $42408$
Product of factors : ${11760}^{40}$
Let $N=11760=2^4\times 3\times 3\times 5\times 7^2$
Then, total number of factors or divisors of $N=(4+1)(1+1)(1+1)(2+1)$
$=5\times 2\times 2\times 3=60$

Sum of all the factors of $N=\frac{2^{4-1}-1}{2-1}\times \frac{3^2-1}{3-1}\times \frac{5^2-1}{5-1}\times \frac{7^3-1}{7-1}$

$=31\times \frac{8}{2}\times 6\times \frac{342}{6}=31\times 4\times 6\times 57=42408$

Product of all factors of $N=(N{)}^{\frac{Totaldivisors}{2}}$

$=(11760{)}^{\frac{60}{2}}=(11760{)}^{30}$