Problem Solving

Any advice for how to approach a question like this? I don't understand how to best approach.

Thank you!

Tom

For more practice, here's a similar problem, along with my solution:

If n and y are positive integers and 450y = n³, which of the following must be an integer?

I. y/(3 x 2² x 5)

II. y/(3² x 2 x 5)

III. y/(3 x 2 x 5²)

a. None
b. I only
c. II only
d. III only
e. I, II, and III

Try to prove that I, II and III DON'T have to be integers.
To this end, plug in the MINIMUM POSSIBLE VALUE for y.

450y = n³ implies that 450y is the cube of an integer.

When we prime-factorize the cube of an integer, we get 3 (or a multiple of 3) of every prime factor:
8 is the cube of an integer because 8 = 2³ = 2*2*2.
27 is the cube of an integer because 27 = 3³ = 3*3*3.

Thus, when we prime-factorize 450y, we need to get AT LEAST 3 of every prime factor.
Here's the prime-factorization of 450y:
450y = 2 * 3² * 5² * y

Since 450 provides only one 2, two 3's, and two 5's, and we need at least 3 of every prime factor, the missing prime factors must be provided by y.
Thus, y must provide at at least two more 2's, one more 3, and one more 5.
Thus, the MINIMUM possible value of y = 2² * 3 * 5.

Plug y = 2² * 3 * 5 into the answer choices:

I. y/(3 x 2² x 5)
(2² * 3 * 5)/(3 x 2² x 5) = 1.
The smallest possible value of y yields an integer.
Eliminate every answer choice that does not include I.
Eliminate A, C and D.

II. y/(3² x 2 x 5)
(2² * 3 * 5)/(3² x 2² x 5) = 1/3.
Not an integer.
Eliminate every remaining answer choice that includes II.
Eliminate E.

The correct answer is B.

Here's another way:

If two integers are equal to each other, they have the same prime factorization. For instance, 2 * 15 = 3 * 10, because each factors as 2 * 3 * 5. This seems obvious, but it'll be useful in a second!

If 375y = x * x, and both are integers, then the same rule applies: each side must have the same prime factorization. The left-hand side is 3*5*5*5*y, so the right hand side must also have be 3*5*5*5*something.

Now notice that the right hand side is a perfect square: x * x. That means we should be able to group it into two identical roots. (For instance, 36 = (2*3) * (2*3).) But if we try to group 375y into two identical roots, here's what we get:

(3 * 5 * 5) * (5 * y)

Yuck! So clearly we're missing a few numbers, but which ones? Well, if he had an extra 3 and an extra 5, we'd have (3 * 5 * 5) * (3 * 5 * 5). (The numbers I just made up are in red.) So (3*5*5)*(3*5*5) could equal x * x.

Since 3*5*5*5*y = x * x = (3*5*5) (3*5*5), notice that y = 15. From there, just plug y = 15 into each of the three options and see which are true.

You might be asking (rightly) why x * x couldn't have been something else, such as (3*5*5*7*11)² or something. It certainly COULD be that (we don't know!) However, since the question asks which of these MUST be an integer, we want to find the minimum possible value of y. Since each of our roots must have a 3 and two 5s in it, the minimum value of x * x = (3*5*5)², so the minimum value of y is 15. Whichever of the three statements is true of the minimum value of y will also be true of every other value of y, since every other value of y will just be a multiple of 15.

Hey,

the question says 375y=x^2
well.. 375= 5X5X5X3
therefore.. y is anything which makes this number a square
lets start from the nearest options, say y= 3X5 which makes the eq.= (5X5X5X3)X(3X5)
therefore from the given options in the question y/30 cannot be an integer as you will need 2 2's to make it a LHS a square.

moving from this argument y/15 and y^2/25 seems likely to result an integer..