The smallest positive integer value of n for which 168 n isa square number

168 = 2*2*2*7*3

so I need to multiply by 2*7*3 or 42
168 * 42 = 7056
sqrt (7056) = 84
so 42 works

Oh, finding the smallest positive integer value of n for which 168n is a square number. That's like searching for a needle in a needle stack, quite a challenge!

Let me put on my clown thinking cap and solve it for you in a fun way.

Since we're looking for a square number, let's start by breaking down 168 into its prime factors. It's like taking apart a puzzle to see what's inside.

168 = 2 * 2 * 2 * 3 * 7

Ah, now that we have the prime factors, we want to make sure that each prime is raised to an even power, so that the result is a perfect square. This is like putting things back together to make a complete picture.

Since the smallest exponent is 1 (from the 3), we need to multiply n by another prime factor to make it an even power. We have two options: 2 or 7.

Let's try the first option, n = 2.

168 * 2 = 336

Oh, the number 336 isn't a perfect square. We're still searching for our needle!

Now, let's try the second option, n = 7.

168 * 7 = 1176

Oh, no! It seems like 1176 is still not a perfect square. Our needle seems to be quite elusive!

Let me put on my detective hat and investigate further. It seems like we need to find a different combination of prime factors to make it work.

After some more clown calculations, it turns out that the smallest positive integer value of n for which 168n is a square number is... *drumroll*... n = 2 * 3 * 7 * 7 = 294!

So, the clownish answer to your question is n = 294. Voila! We caught that sneaky needle at last!

To find the smallest positive integer value of n for which 168n is a square number, we need to determine the factors of 168.

The prime factorization of 168 is:
168 = 2^3 × 3 × 7

For a number to be a perfect square, each prime factor must appear an even number of times in its prime factorization.

The prime factorization of a square number has only even powers of primes. In this case, since 168 already has a power of 3, we need to ensure that the exponents of 2 and 7 are even.

Looking at the powers of 2, the smallest exponent that will make it even is 2.
2^2 = 4

If we include 2^2 in the prime factorization of 168n, the remaining factors are 3 and 7. Since the exponents of both 3 and 7 are odd, we need to find the LCM (Least Common Multiple) of the remaining factors.

LCM(3, 7) = 3 × 7 = 21

Therefore, the smallest positive integer value of n for which 168n is a square number is:
n = LCM(3, 7) = 21

To find the smallest positive integer value of n for which 168 * n is a square number, we need to analyze the prime factorization of 168.

The prime factorization of 168 is: 2 * 2 * 2 * 3 * 7.

To make 168 * n a perfect square, we need to have an equal number of each prime factor in the prime factorization of n. Since we already have three 2's, one 3, and one 7 in the prime factorization of 168, to make the product a perfect square, we need to include an additional two 2's, one 3, and one 7 in the prime factorization of n.

Therefore, the smallest value of n that makes 168 * n a perfect square is: n = 2 * 2 * 2 * 3 * 7 = 168.

Thus, the smallest positive integer value of n for which 168 * n is a square number is 168.