manhattan test series question

[manikdhingra]

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?
a.5
b.6
c.15
d.16
e.18

[Shooter]

We need to make the length maximum. They have given that prime numbers can be distinct.

so we will start with least prime '2' and see that we use only this number. If we use distinct numbers we will get the length less than maximum.

x+3y<1000

Pick the numbers and make sure that the condition satisfies:

2^9=512 (X)
2^7 =128*3 = 348 (Y)

2^9+3*2^7 < 1000

Try next value:

2^10+3*2^7>1000 (condition is not satisfying)

=> 16 is the maximum length.

[Makumajon]

x+3y<1000

To get the maximum length of x or y, you must try to make these integers using 2 as factor as many as possible, because 2 is the lowest prime and hence using it will increase the length more than using any other prime.

Now look again x+3y<1000. Of course, you will try to maximize the value of x and minimize the value of y (because y has already the multiple 3, 3y will increase quickly as y increases, but without effectively increaseing the length).

So, raise x so that it becomes quite close to 1000.
Suppose x=2^10=1024 (not possible).
Suppose, x=2^9=512 (choose it).

So, 3y<1000-512
3y<488
y<162.67

Say, y=2^7=128 (possible)
y=2^8=256 (impossible).

So, x's length is 9, while y's length is 7. Answer 9+7=16

[GmatG]

Originally Posted by Makumajon

x+3y<1000

To get the maximum length of x or y, you must try to make these integers using 2 as factor as many as possible, because 2 is the lowest prime and hence using it will increase the length more than using any other prime.

Now look again x+3y<1000. Of course, you will try to maximize the value of x and minimize the value of y (because y has already the multiple 3, 3y will increase quickly as y increases, but without effectively increaseing the length).

So, raise x so that it becomes quite close to 1000.
Suppose x=2^10=1024 (not possible).
Suppose, x=2^9=512 (choose it).

So, 3y<1000-512
3y<488
y<162.67

Say, y=2^7=128 (possible)
y=2^8=256 (impossible).

So, x's length is 9, while y's length is 7. Answer 9+7=16

Great Explanation Makumajon.