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Divisibility by 3

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by mdavidm_531 » Tue May 08, 2012 8:05 pm
AJ1440 wrote:If 144/n^2 is an integer is n divisible by 3?
if n = 2, 144/2^2 = 144/4 = 36 (integer); n is not divisible by 3

if n = 3, 144/3^2 = 144/9 = 16 (integer); n is divisible by 3

By counter-example, we conclude that we can't say.

Moreover,

144 = (2^4)(3^2)
144/n = (2^4)(3^2)/n = integer; n could be any power of 2 or any power of 3
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by zueswoods » Wed May 09, 2012 8:43 am
If we break down 144 into factors

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

note: in order for a number to be a divisor, all the factors of the divisor must cancel out with the factors of the numerator

If n is a divisor of 144 it means n must have at least a 2 (or two 2's) or a 3 as its factors

n could = 2 -----> n^2 = 2^2
n could = 2^2 -----> n^2 = 2^4

or

n could = 3 -----> n^2 = 3^2

or

n could = (3)(2) ----- n^2 = 3^2 * 2^2


If n only has 2 in its factors, it wont be divisible by 3, however if n has a 3 it will be.
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