If $y$ is a positive integer, for how many different values of $y$ is $\sqrt[3]{\frac{144}{y}}$ a whole number?
Answer (2)
There are 2 different values of y for which 3 y 144 is a whole number: 144 and 18. This occurs because these values lead to perfect cubes as the result when dividing 144.
;
Find x such that x = 3 y 144 is a whole number, implying y = x 3 144 .
Determine the prime factorization of 144 = 2 4 ⋅ 3 2 .
Identify possible values of x 3 that divide 144 and are perfect cubes: x 3 = 1 and x 3 = 8 .
Calculate the corresponding y values: y = 144 and y = 18 , resulting in 2 different values of y .
Explanation
Problem Analysis We are given that y is a positive integer and we need to find how many different values of y make 3 y 144 a whole number. Let's analyze the problem and devise a plan to solve it.
Expressing y in terms of x Let x = 3 y 144 , where x is a whole number. Then, cubing both sides, we get x 3 = y 144 . Therefore, y = x 3 144 . Since y must be a positive integer, x 3 must be a divisor of 144 .
Prime Factorization of 144 Now, let's find the prime factorization of 144 . We have 144 = 1 2 2 = ( 2 2 ⋅ 3 ) 2 = 2 4 ⋅ 3 2 . So, 144 = 2 4 ⋅ 3 2 .
Possible values of x^3 Since x 3 is a divisor of 144 , we can write x 3 in the form 2 a ⋅ 3 b , where 0 ≤ a ≤ 4 and 0 ≤ b ≤ 2 . Also, since x 3 is a perfect cube, the exponents a and b must be multiples of 3 . Therefore, the possible values for a are 0 and 3 , and the only possible value for b is 0 .
Calculating Possible x^3 values Thus, the possible values of x 3 are:
2 0 ⋅ 3 0 = 1
2 3 ⋅ 3 0 = 8
Calculating y values Now, we find the corresponding values of y :
If x 3 = 1 , then y = 1 144 = 144 .
If x 3 = 8 , then y = 8 144 = 18 .
Final Answer Therefore, the possible values of y are 144 and 18 . There are two different values of y that satisfy the given condition.
Conclusion Thus, there are 2 different values of y for which 3 y 144 is a whole number.
Examples
Understanding how numbers divide into perfect cubes is useful in various fields. For example, in cryptography, certain encryption methods rely on the difficulty of finding cube roots modulo a large number. Also, in engineering, when designing structures or systems, it's often necessary to ensure that certain ratios result in whole numbers to avoid instability or resonance. For instance, if you're designing a speaker enclosure, the ratio of the volume to a certain parameter might need to be a perfect cube for optimal sound quality. This problem demonstrates a basic principle of number theory that can be applied in more complex scenarios.
