Question

Prove that 33! is divisible by 2(15). What is the largest value of n so that 33! is divisible by 2(n) ?

Answer 1

33!=1*2*3*4.....*33

33!=(1*3*5*7......*33)*(2*4*6*8*10......*32)

let(1*3*5*7.....*33)=a

33!=a*(2*4*6*8*10......*32)

33!=a*(6*10*12*14*18*20*22*24*26*28*30)*(2*4*8*16*32)

know that 2*4*8*16*32=2^15,

(6*10*12*14*18*20*22*24*26*28*30)=(3*5*6*7*9*10*11*12*13*14*15)*(2^11)

33!=a*(3*5*6*7*9*10*11*12*13*14*15)*(2^11)(2^15)

(3*5*6*7*9*10*11*12*13*14*15)=(3*5*3*7*9*5*11*3*13*7*15)*(2^5)

33!=a*(3*5*3*7*9*5*11*3*13*7*15)*(2^5)(2^26)

33!=a*(3*5*3*7*9*5*11*3*13*7*15)*(2^31)

so n=31