If x is a positive integer and x^2 is divisible by 32, then

Imo C

32=2^5
nearest squre =2^6=8^2
hence x=8 and the lagest int dividing 8 is 8

Thank you for solving. However I am confused why not x is divisible by16.

if x is 16 then x^2 is 256 which is divisible by 32. Appreciate your help.

Thank you and it makes sense.

Very interesting question.

Here is the similar problem link:
if-n-is-a-positive-integer-and-n-2-is-divisible-by-72-then-129929-20.html#p1341433

I got one of the questions wrong and then got another right, I think some problem has been discussed in the above link on the first page.

Hope it helps.

Hello Bunuel

"The largest positive integer that must divide x, means for lowest value of x which satisfies the given statement in the stem."

Why are we looking for the smallest value of x ?

Simply put x = 8 and get the answer. If you put a number greater than 8, then you would have less options to eliminate and more to check.

Hi Bunnel,

What if the question is as below:

If P is a positive integer and p^3 is divisible by 144, then the largest positive integer that must divide p is

How do we calculate using the "k" method when the cubeth root is asked for.

Hi Pretz,

In the original question, we're essentially looking for the smallest multiple of 32 that is a perfect square....

32(2) = 64......which is 8^2.

Using prime factorization, we know that 32K = (2^5)K

By making K = 2, we have (2^5)(2) = 2^6

We can then break 2^6 into 2 equal "pieces": (2^3)(2^3) which equals (8)(8)

To answer your question, we're going to use a similar approach:

We're looking for the smallest multiple of 144 that is a perfect cube....

144K = (2^4)(3^2)K

By making K = 12, we have (2^4)(3^2)(2^2)(3) = (2^6)(3^3)

We can break this down into 3 equal "pieces": [(2^2)(3)][(2^2)(3)][(2^2)(3)] = (12)(12)(12)

GMAT assassins aren't born, they're made,
Rich

Bunuel
If (F) = 128 was one of the possible answers, would that have been the correct choice?
Im trying to understand the concept behind this kind of exercise.

So 128 = 2^7
Because x^2 we have to get a perfect cube (with highest possible number).
If k = 2^7 we get (32 + 128) 2^12 or 4^3 * 4^3.

Is this logic correct?

maybe a dumb question: The largest positive integer that must divide x, means for lowest value of x which satisfies the given statement in the stem -- why?

Why is it not the largest value of x. For example, if x = 100 why cant x be 100 which will divide 100 and is the largest positive integer. What am I missing?

We don't need to get perfect cube but the perfect pair of square.
Like in Question, X^2 is divisible by 32
The minimum value of x^2 can be 32 but 32 is not a perfect squre. 2^5 need another 2 to make it perfect square. So the least value of X^2 will be 64 {2^6 = (2^3)^2}

And if you say X^2 is divisible by 128 = 2^7
We need perfect square (every even power is one) so multiply 128 with 2
Giving 256 = (2^4)^2
So the maximum number by which x can be divided is 2^4 = 16.
Hope you get it.

Posted from my mobile device

We can say that the largest divisor of a number is its least value (the least value of a prime number:2 and it's largest divisor is 2 only, but we can't say the maximum value part for such number which can change. The number has to be fixed to know it's divisor)

In question the least value Of X^2 is 64 (because 32,16,8,4,2 are either not divisible by 32 or is not a perfect square)
That's why the least value of X is 8. Now 8 is divisible by 4 factors (1, 2,4&8) where 8 is maximum.
It's not the largest value of X for which we want the divisor because the largest value of x^2 can extend to infinity which won't have a single value.

In your example 100 is divisible by 100 but the 100 must be because of some constraint. Let say x^2 is divisible by 1000 so x^2 has to be a multiple of 1000, we also know x^2 is a perfect square while 1000 is not that's why we will multiply 1000 with 10
Giving 10,000 as perfect square = X^2
And X = 100
Whose maximum divisor is 100.

Posted from my mobile device

The largest positive integer that must divide , means for lowest value of which satisfies the given statement in the stem.
x where is an integer (as is positive).

--> , as is an integer , also must be an integer. The lowest value of , for which is an integer is when --> -->

Posted from my mobile device

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