Integers Multiplication Question

Question

The even positive integers are multiplied together, as in 2 x 4 x 6 x ... x n, where n is some even integer. What is the smallest value of n for which this product is divisible by 2016?

Answer 1

write 2016 as a product of prime factors

2016=2*2*2*2*2*3*3*7

you need five 2's, two 3's and a 7 as factors in the product

the 2's are no problem because each even number has at least one 2 in it

the 3's are no problem because 6 has a 3 and 12 has a 3

you now need to get to an even number that has a 7 in it; that would be 14

2*4*6*8*10*12*14=645,120

645,120/2016=320

the smallest value of n is 14