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 ${}^n{P_4} : {}^n{P_5} = 1 : 2$,then $n = $

The number of positive integral solutions of the equation $xyz = 3000$ is:

The number of four-digit natural numbers which contain exactly two distinct digits is:

If in a regular polygon the number of diagonals is $54$,then the number of sides of this polygon is

The value of $^{15}C_3 + ^{15}C_{13}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module