Question

Find the smallest value of k, when 264k is a perfect square

Answer 1

okay 1st write the prime factorization of 264
@Love.Study :)

Answer 2

I did

Answer 3

@imqwerty

Answer 4

okay what did you get

Answer 5

264

Answer 6

@imqwerty

Answer 7

i mean what is the prime factorization of 264

for example prime factorization of 12 is->
\(12=2 \times 2 \times 3\)
:) like this

Answer 8

cant you just divide by 4 because a square as 4 sides and its suppose to be all even so take 264 and divide 4 to get your answer

Answer 9

@annie12m we have to multiply
and k prolly has to be a positive integer so that this problem becomes legit

Answer 10

Oh, 2 multiplied by 2 multiplied by 2 multiplied by 2 multiplied by 3 multiplied by 11 @imqwerty

Answer 11

its almost correct
you just gotta remove one 2

Answer 12

correct prime factorization of 264 is this-
\(264=2 \times 2 \times 2 \times 3 \times 11\)

Answer 13

we can write it like this-
\(264=2^3 \times 3 \times 11\)

Answer 14

Yes I wrote that @imqwerty

Answer 15

you wrote one extra 2
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Answer 16

I wrote it accidentally in the message, in the copy I've written three 2, one three and one 11 @imqwerty

Answer 17

to be a prime the power of each term in the prime factorization must be even

but here we see that-> \(2^{\color{red}3} \times 3^{\color{red}1} \times 11^{\color{red}1} \)
so all powers are odd
so k must be a number which on multiplication with 264 makes all those powers even

Answer 18

its alright :)

Answer 19

to make all powers as even
we go step by step-

1st we gotta make \(2^3\) to have an even power
so we just multiply it by \(\color{red} 2\) to get this-> \(2^3 \times 2=2^4\)

2nd we make \(3^1\) to have even power
so we just multiply it by \(\color{red}3\) to get this- \(3^1 \times \color{red}3=3^3\)

similarly we multiply \(11^1\) by \(\color{red}{11}\)

Answer 20

so overall we are multiply k
and k is nothing but the extra numbers multiplied to make the powers even :)

so \(k= 2 \times 3 \times 11 = 66\)

Answer 21

Thank you @imqwerty

Answer 22

np :)