The problem asks us to find the smallest number 'y' such that the product of 'y' and 162 results in a perfect cube.

Understanding Perfect Cubes

A perfect cube is a number that can be obtained by multiplying an integer by itself three times. In terms of prime factorization, a number is a perfect cube if all the exponents in its prime factorization are multiples of 3. For example, $27 = 3^3$ and $64 = 4^3 = (2^2)^3 = 2^6$.

Prime Factorization of 162

First, let's find the prime factorization of the given number, 162:

• $162 \div 2 = 81$
• $81 \div 3 = 27$
• $27 \div 3 = 9$
• $9 \div 3 = 3$
• $3 \div 3 = 1$

So, the prime factorization of 162 is $2 \times 3 \times 3 \times 3 \times 3$, which can be written as:

$$162 = 2^1 \times 3^4$$

Finding the Smallest Number y

We want to find the smallest number 'y' such that $y \times 162$ is a perfect cube. Let the prime factorization of 'y' be $2^a \times 3^b$.

The product is: $$ y \times 162 = (2^a \times 3^b) \times (2^1 \times 3^4) = 2^{a+1} \times 3^{b+4} $$

For this product to be a perfect cube, the exponents $(a+1)$ and $(b+4)$ must be the smallest possible multiples of 3.

• Exponent of 2: We need $a+1$ to be a multiple of 3. The smallest multiple of 3 that is greater than or equal to 1 is 3. So, we set $a+1 = 3$, which gives $a = 3 - 1 = 2$.
• Exponent of 3: We need $b+4$ to be a multiple of 3. The smallest multiple of 3 that is greater than or equal to 4 is 6. So, we set $b+4 = 6$, which gives $b = 6 - 4 = 2$.

To find the smallest number 'y', we use the smallest required exponents for the prime factors.

Therefore, the smallest value for 'y' is:

$$ y = 2^a \times 3^b = 2^2 \times 3^2 $$

Calculating the value of 'y':

$$ y = 4 \times 9 = 36 $$

Verification

Let's check if multiplying 162 by 36 results in a perfect cube:

$$ 36 \times 162 = (2^2 \times 3^2) \times (2^1 \times 3^4) $$

Combine the powers of the same base:

$$ = 2^{2+1} \times 3^{2+4} = 2^3 \times 3^6 $$

Now, let's see if $2^3 \times 3^6$ is a perfect cube. We can rewrite it as:

$$ = (2^1)^3 \times (3^2)^3 = (2 \times 3^2)^3 $$

$$ = (2 \times 9)^3 = 18^3 $$

Since $36 \times 162 = 18^3 = 5832$, which is a perfect cube, our value for 'y' is correct.

Conclusion

The smallest number 'y' such that $y \times 162$ is a perfect cube is 36.