Number properties

[Nigogo]

For any integer k > 1, the term "length of an integer" refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?
5
6
15
16
18

[xcusemeplz2009]

IMO D

2^9+3*2^7

[Nigogo]

so you mean the unswer is D 16? Can you plz explain how did u get it?

[xcusemeplz2009]

sorry i made a mistake tough ans is same but the no. is diff.

we need to find two number to satisfy the eqn. X+3Y>1000

and we need to find the maximum length (number of prime factors not necessarily distinct)

since maximum therefor better to take minimum positive prime no. which is 2

now ifx=2^9=512
then 3y has to be <1000-512=482
for y we can have max no. as 2^7( for more higher power of 2 the no. will exceed 1000)
so x=2^9(tot length 9)
and Y=2^7
(tot length 7)

adding ans 16

[Nigogo]

Thank you :!: