Perfect Square Product

The prime factorization of $792 = 3^2 * 2^3 * 11$ , or in more useful terms, $3^2 * 2^2 * 2 * 11$ . Since the $3^2$ and $2^2$ are already perfect squares, we just need to multiply by something with another $2$ and another $11$ in it. $22$ is an obvious choice, and is the first solution. Any other multiple of $22$ with a perfect square would work as well. So $22*4, 22*9, 22*16, 22*25$ , and $22*36$ are also solutions. $22*49$ is larger than $1000$ , so we must stop with $36$ .

The solutions are $22, 88, 198, 352, 550$ , and $792$ for a total of $6$ .

From the question, we can say that any positive integer multiplied with $792$ must yield a perfect square.

i.e. Let's say $x \times 792$ is a perfect square.

Find all the factors of 792, we get

$x \times 4 \times 9 \times 11 \times 2$ must be a perfect square.

Here, $4$ and $9$ are already squares, but $11$ and $2$ are not. So, in order that the above quantity is a perfect square, $x$ must contain the term $11 \times 2$ in it.

So, $x$ contains $11 \times 2 \times$ something . Note that something must be a perfect square.

So, we put different squares starting from 1, 4, 9, 16, ... and calculate $11 \times 2 \times 1$ , $11 \times 2 \times 4$ ..... This quantity must be less than 1000.

We see that $11 \times 2 \times 1$ , $11 \times 2 \times 4$ , .... $11 \times 2 \times 36$ are all less than 1000. But $11 \times 2 \times 49$ is not.

Hence, there are $6$ possible values.

That's the answer!

The prime factorization of $792$ is $2^3 \cdot 3^2 \cdot 11$ . A perfect square is a number where all of the exponents of all prime factors is even. Therefore, the positive integers that satisfy is in the form $22k^2$ , where $k$ is any positive integer. The range that $k$ can be in in order for $22k^2$ to be less than $1000$ is $[1,6]$ , so there are $\boxed{6}$ possible integers.

792 factors as 2 x 2 x 2 x 3 x 3 x 11.

we strictly need a "2" and a "11" to make it a perfect square.

so the simplest, we can multiply " 22 " to 792 for makeing a perfect square.

" a perfect square x 22 " x 792. is a perfect square.

so we can multiply 22 with 1,4,9,…. and the number should be less than 1000.

that happens with only upto 36.

so the numbers are 22 x 1, 22 x 4, 22 x 9, 22 x 16, 22 x 25 and 22 x 36.

there are 6 numbers that we want

Well, $792 = 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 11$ . Then, any number multiplying this needs to multiply $2 \cdot 11$ , because we need to have an even quantity of each prime factor. Then, $2 \cdot 11$ needs to be multiplied by an even quantity of factors.

Note that $2 \cdot 11 \cdot 7 \cdot 7 = 1078$ . Hence, the prime factors that we can add up are $2$ , $3$ and/or $5$ . Then, note that $2\cdot11\cdot5\cdot5 = 550$ . Then, the pair of $5$ 's will not be accompanied by another pair (because then the number will be greater than $1000$ ). We continue with the $3$ 's. $2\cdot11\cdot3\cdot3 = 198$ . In this case we can only add up a pair of $2$ 's. Then, with the $2$ 's, we have $2\cdot11\cdot2\cdot2$ and $2\cdot11\cdot2\cdot2\cdot2\cdot2$ . So, let's list below, and formally, all the possible numbers:

$2\cdot11$

$2\cdot11\cdot2\cdot2$

$2\cdot11\cdot2\cdot2\cdot3\cdot3$

$2\cdot11\cdot2\cdot2\cdot2\cdot2$

$2\cdot11\cdot3\cdot3$

$2\cdot11\cdot5\cdot5$

Finally, the answer is $\boxed {6}$ .

792 can be written as 2^3 * 3^2 * 11, so first requirement is to get the already present factors in 792 to be perfect square, so the least number which is required is 2*11 i.e. 22, now we know that only multiple of 22 can satisfy the condition, so apart from 22 other multiples should be having the perfect square terms, so 22 * 4 , 22 * 9, 22 * 16, 22 * 25, 22 * 36. after this the number will be greater than 1000, so correct answer is 6..

First, take note that $792 = 8*9*11 = 2^{3}*3^{2}*11$ . Thus, the smallest number we must have to multiply by 792 in order to produce a square number is $2*11 = 22$ . Next, we must find the multiples of 22 which are of the form $22*n^{2}$ . Noticing that $22*45 = 990$ and $22*46 = 1012$ , $n^{2}$ must then be less than or equal 45. Thus, $n$ can assume one of the following values: (1, 2, 3, 4, 5, 6). Numbers from 7 onward are ruled out since $7^{2} = 49 > 45$ , thus we only have 6 numbers that satisfy the problem.

La decomposition en facteur premier de 792 est 2^3 3^2 11 , pour qu'un carree soit parfait il faut que touts ses nombres premiers (decomposeurs) aillent soit des carrees donc ayant des puissances paires ainsi on remarque que pour que 792 soit carree il faut le multiplier par 2 11 (en le multipliant par ce nombre toutes les puissances des nombres premiers deviennent paires) , et le nombre des multiples par carrees (cad 22 multiplie par 1, 4, 9...) inferieurs a 1000 est le nombre cherché. 22 (6^2) = 792 et 22*(7^2) = 1078. Donc le nombre d'entiers positifs qui sont inferieurs a 1000 et representent un carree l'orsqu'on les mutiplie par 792 est \boxed{6}.

$792$ can be factorized into $2^3.3^2.11$ . So, the smallest positive integer which yields a perfect square number when multiplied by $792$ is $22$ . Since $792*22$ is a perfect square number, when multiplied to any perfect square number, it will remain a perfect square number. They are $22*1, 22*4, 22*9, 22*16, 22*25$ and $22*36$ . $\boxed{6}$ is the answer.

factorize 792. By factorizing 792 we get the following : 2x2 x 3x3 x 2x11. now if we multiply 2X11 that is 22 with 792 we get: 2x2 x 3X3 x 2X11 x 2x11 which is a perfect square .. Now to get more perfect squares we can go on multiplying perfect squares with this no..like 4,9,16... taking care that 2x11x(the perfect square) doesn't exceed 1000.. so up to how many squares can we multiply?? divide 1000 by 22, we get 45.45.. so 22X45 is just below 1000, multiplying anything above 45 with 22 will take the product over 1000.. so we can multiply perfect squares upto 45 with 22 and multiply that product with 792..

so the numbers are: 22 X 1=22 ( this is in fact the first number i have calculated) 22 X 4=88 22 X 9=198 22 X 16=352 22 X 25=550 22 X 36=792...

We must know if 792 = 2^3 . 3^2 . 11, So, positive integers less than 1000 yields a perfect square number when multiplied by 792 are 2^1.11 , 2^3 . 11 , 2^5 . 11 , 3^2 . 11 , 3^4.11 , and 2^1 . 3^2 . 11. There are 6 positive integers. Answer : 6