arun sharma

Question

please explain question 23 and 24 

Answer 1

23) 144 = 2⁴ * 3²

     sum of divisor of 144 = (2⁰+2¹+2²+2³ +2⁴) * (3⁰+3¹+3²)

     =(2⁵-1)/(2-1) * (3³-1)/(3-1) = 31*13 = 403

     160 = 2⁵ * 5¹

     sum of divisor of 160 = (2⁰+2¹+2²+2³ +2⁴ +2⁵) * (5⁰+5¹)

     =(2⁶-1)/(2-1) * (5²-1)/(5-1) = 63*6 = 378

now SUM of SUM OF DIVISOR = 403 + 378 =781

24)

    96 = 2⁵ * 3¹

sum of EVEN divisor of 96 = (2¹ + 2² + 2³ + 2⁴ +2⁵) * (3⁰ +3¹)

(we are not adding 2⁰  term becoz when we multiply with any odd term we get odd term only )

=2(2⁵-1)/(2-1) * (3²-1)/(3-1) = 62 * 4 = 248

3600 = 2⁴ * 3² * 5²

sum of ODD divisor of 96 = (3⁰ +3¹ +3²) * (5⁰+5¹+5²)

(here for odd divisor we consider only odd term as we know   even * odd =even)

= (3³-1)/(3-1) * (5³-1)/(5-1)  =13 * 31 = 403

now SUM is 248 + 403 = 651