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CBSE Class 10 Answered

Let P(n) = (n+1)(n+3)(n+5)(n+7)(n+9). What is the largest integer that is a divisor of p (n) for all positive even integers n?
Asked by vikasg13.hardware | 17 Jun, 2018, 10:54: AM
answered-by-expert Expert Answer
for n being 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 will be 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.
Answered by Arun | 18 Jun, 2018, 11:47: AM

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