Question

What is the largest integer that is a divisor of (p+1)(p+3)(p+5)(p+7)(p+9) for all positive integer p?

Answer 1

Answer:

HELLO DEAR

Step-by-step explanation:

For n be odd (2k-1):

p(n)=2k(2k+2)(2k+4)(2k+6)(2k+8)

       =32*k(k+1)(k+2)(k+3)(k+4) ,

       =32*5 consecutive numbers ,

which divisible by 32*(5!)=3840

for n being even (2k)

p(n)=(2k+1)(2k+3)(2k+5)(2k+7)(2k+9)

         at least 1 of the terms in the product will be divisible by 3 and at least one by 5

hence it will be divisible by 15.

Therefore, p(n) will be divisible by 15 for all positive n.