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If n is a positive integer and n^2 is divisible by 72, then [#permalink]

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09 Jan 2007, 00:21

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2. If n is a positive integer and n^2 is divisible by 72, then the largest possible positive integer that must divide n is
A) 6
B) 12
C) 24
D) 36
E) 48

I get E on this one. Its a pain because I started with 12, the easy answer. 12 square is 144/72=2. 36 works but 48 square is 2304/72 is 32. Hence E. I dont know my squares that high, took a minute to do it longhand. Hope not to see it on the test.

E for me. This one took me a bit of time to figure out.

n^2 is divisible by 72

Prime Factors of 72 are 2^3*3^2

Hence n^2 = 2^2*3^2*2*k where k is any integer such that 2*k is a square
Let 2*k=j
Therefor the largest number that n is divisible by is 2*3*j=6*j i.e. some multiple of 6

Given the choices the bigggest multiple of 6 is 48 (6*8)i.e. 2*k =64

48^2 is divisible by 72. Hence the largest integer that divides n is 48.
_________________

2. If n is a positive integer and n^2 is divisible by 72, then the largest possible positive integer that must divide n is A) 6 B) 12 C) 24 D) 36 E) 48

Surely, of the choices given, 48 is the largest # that could divide n, but 12 is the one that must divide n.

The wording is a bit tricky, watch out for must/could/cannot/etc.

Last edited by Andr359 on 09 Jan 2007, 21:40, edited 1 time in total.

My vote is for B. My approach was the same as Fig's.

n should be a multiple of 12. For any value of n, 12 should divide n. The other nos. MAY or MAY not depending on n.

For example if n=36 then n^2 is divisible by 72, n will not be divisible by 48 but will be divisible by 12. The question asks for the largest number that MUST divide n. So 12 should be the answer.

n^2 = ( (3)^2 * (2)^2 * 2 ) * k^2 => n = 3*2*sqrt(2)*k : non sense, as n is an integer, it forces k to not be an integer.

We must have an integer j such that : n^2 = ( (3)^2 * (2)^2 * 2 ) * 2 * j^2 <=> n = 3*2*2 * j <=> n = 12 * j

Fig, could you please explain the bolded sections in a little more detail.
For the first one, since n^2 is divisible by 72, shouldn't it be written as
n^2 = ( (3)^2 * (2)^2 * 2 ) * k : (where k is any integer)
The second one, I'm totally lost

n^2=72k=36*2k = 6^2*2k
2k needs to be the square of a number
k=2m^2, where m is a positive integer
therefore
n=12m

Therefore 12 is the largest possible number that must divide n. Other possible numbers that must divide n would be 1,2,3,4,6. Other possible numbers that may divide n could be anything, depend on what m is.
_________________

Keep on asking, and it will be given you;
keep on seeking, and you will find;
keep on knocking, and it will be opened to you.

n^2=72k=36*2k = 6^2*2k 2k needs to be the square of a number k=2m^2, where m is a positive integer therefore n=12m

Therefore 12 is the largest possible number that must divide n. Other possible numbers that must divide n would be 1,2,3,4,6. Other possible numbers that may divide n could be anything, depend on what m is.

2. If n is a positive integer and n^2 is divisible by 72, then the largest possible positive integer that must divide n is A) 6 B) 12 C) 24 D) 36 E) 48

OA I have is B.

but for me the question is not properly worded as it says largest possible. is 12 the largest possible value that divide n? lets see....

(n^2)/72 = k
n^2 = 72k
n = 6 x sqrt(2k)

now, k could be a minimum of 2 or 2x2x2, 2x3x3, 2x4x4, 2x5x5, 2x6x6 or 2(n^2).

If k = 2, n = 12 and n is divisible by 12.
If k = 2x2x2, n = 24 and n is divisible by 24.
If k = 2x3x3, n = 36 and n is divisible by 36.
If k = 2x4x4, n = 48 and n is divisible by 48.
If k = 2x5x5, n = 60, and n is divisible by 60.

so we know that, n is a multiple of 12 and it is divisible by any number that is a multiple of 12.

If n is a positive integer and n^2 is divisible by 72, then the largest possible positive integer that must divide n is A) 6 B) 12 C) 24 D) 36 E) 48

The problem says the largest possible that must divide n. LetÂ´s solve step by step.
1) n^2 is multiple of 72 => n^2 = 72 * a (a = integer>=1)
2) n^2 = 2^3 * 3^2 * a = 2^2 * 3^2 * 2a
3) n = 2 * 3 * sqrt(2a)
4) n is integer => sqrt(2a) must be integer too. LetÂ´s say that sqrt(2a) = 2*k, k an integer. sqrt(2a) has to be even because an integer sqrt of an even number is always even as well (think 16 and 4, 36 and 6, 144 and 12, etc).
5) n = 2 * 3 * 2k = 4 * 3 * k = 12 * k. k could be 1, 5, 24, 8239823698298, any integer.
6) What is the largest +ve integer that must divide n?
7) Would it be 48? What if k = 5? n = 12 * 5 = 60; can 48 divide 60? No.
8) Which numbers must divide n = 12 * k? 2, 3, 4, 6, and 12.
The trick is to focus 1st on the "must", and only afterwards on the "largest".