Question

Find the smallest positive integer with which one has to divide 336 to get a perfect square.

Answer 1

Hint: First we have to define what the terms we need to solve in the problem are.
Since we just need to know such things about the perfect numbers, which is the numbers that obtain by multiplying any whole numbers (zero to infinity) twice, or the square of the given numbers yields a whole number like $\sqrt {25} = 5$ or $25 = {5^2}$

Complete step-by-step solution:
Let we need to find the number which divides \[336\] to get the perfect square, so first, rewrite the given number into the form of square root separation and then we can find the perfect square;
Thus \[336\] can be written as \[336 = 4 \times 4 \times 3 \times 7\] (if we multiply, we get the same result as \[336\]); after separating now we need to eliminate the non-perfect square numbers. since $4$ is a perfect square by the perfect square definition $4 = {2^2}$ or $\sqrt 4 = 2$ hence eliminate the numbers $4$ which occur both times.
After eliminating four now we only left with $3$ and $7$ which are the non-perfect square numbers because $3$ cannot be written as in the form of square root and also $7$ is the non-perfect square number for the same reason, thus the non-perfect square numbers for \[336\] are $3$ and $7$. now we will divide the number \[336\] by $3$ and $7$ get the least positive integer which will divide to make it a perfect square. Thus $\dfrac{{336}}{{3 \times 7}}$we get $\dfrac{{336}}{{3 \times 7}} = \dfrac{{112}}{7}$ (divided by three) and further solving we get $\dfrac{{112}}{7} = 16$ (is the remainder)
Clear $16$ is the perfect square because $16 = {4^2}$ or $\sqrt {16} = 4$; hence we divide $21$ (seven into three) to get the perfect square for the number \[336\]. Thus, the smallest positive integer is $21$

Note: Since $2,3,4,6,7$ all numbers can divide the number \[336\], but after dividing we only need the perfect square number like $16$ as a remainder so hence $21$ is the smallest positive integer which divides the given number and make it a perfect square..