The sum of the divisors of 2^5 cdot 3^4 cdot | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Any divisor of $2^5 \cdot 3^4 \cdot 5^2$ is of the form $2^a \cdot 3^b \cdot 5^c$ where $0 \le a \le 5$,$0 \le b \le 4$,and $0 \le c \le 2$.The su

Vedclass · Accepted Answer

$3^2 \cdot 7^1 \cdot 11^2 \cdot 31$

Vedclass · Answer

(B) Any divisor of $2^5 \cdot 3^4 \cdot 5^2$ is of the form $2^a \cdot 3^b \cdot 5^c$ where $0 \le a \le 5$,$0 \le b \le 4$,and $0 \le c \le 2$.The sum of the divisors is given by the product of the geometric series for each prime factor:Sum $= (1 + 2 + 2^2 + 2^3 + 2^4 + 2^5)(1 + 3 + 3^2 + 3^3 + 3^4)(1 + 5 + 5^2)$Using the formula for the sum of a geometric series $\frac{r^n - 1}{r - 1}$:Sum $= \left( \frac{2^6 - 1}{2 - 1} ight) \left( \frac{3^5 - 1}{3 - 1} ight) \left( \frac{5^3 - 1}{5 - 1} ight)$Sum $= (64 - 1) 	imes \left( \frac{243 - 1}{2} ight) 	imes \left( \frac{125 - 1}{4} ight)$Sum $= 63 	imes \frac{242}{2} 	imes \frac{124}{4}$Sum $= 63 	imes 121 	imes 31$Since $63 = 9 	imes 7 = 3^2 	imes 7$ and $121 = 11^2$,the sum is $3^2 \cdot 7^1 \cdot 11^2 \cdot 31$.

The sum of the divisors of $2^5 \cdot 3^4 \cdot 5^2$ is

Similar Questions

If ${ }^{2 n} C_{r} = { }^{2 n} C_{r+2}$,find $r$.

All the five-digit numbers in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The $97^{th}$ number in the list does not contain the digit:

How many numbers greater than a million can be formed with the digits $2, 3, 0, 3, 4, 2, 3$?

Eight chairs are numbered $1$ to $8$. Two women and three men wish to occupy one chair each. First,the women choose the chairs from amongst the chairs marked $1$ to $4$,and then men select the chairs from amongst the remaining. The number of possible arrangements is

If $n = ^mC_2,$ then the value of $^nC_2$ is given by

Vedclass Test Series

Exam Paper Generator

Online Exam Module