Question
The variable c represents a whole number between 1 and 100. The values of the expressions c^1/2 and c^2/3 are both whole numbers for only one value of c. What whole number does c represent?
Answers
Answered by
Bot
We can start by writing out the prime factorization of c. Let's assume that c has a prime factorization of the form:
c = p1^a1 * p2^a2 * ... * pn^an
where p1, p2, ..., pn are distinct primes and a1, a2, ..., an are positive integers.
Then, we can rewrite c^1/2 and c^2/3 as:
c^1/2 = p1^(a1/2) * p2^(a2/2) * ... * pn^(an/2)
c^2/3 = p1^(2a1/3) * p2^(2a2/3) * ... * pn^(2an/3)
For c^1/2 and c^2/3 to be whole numbers, all of the exponents must be integers. This means that a1, a2, ..., an must be multiples of 2 and 3, respectively. In other words, the prime factorization of c must be of the form:
c = p1^(2x) * p2^(2y) * ... * pn^(2z)
where x, y, z are positive integers.
Since c is between 1 and 100, the only possible values of c that satisfy this condition are:
c = 2^2 * 5^2 = 100
Therefore, the whole number c represents is 100.
c = p1^a1 * p2^a2 * ... * pn^an
where p1, p2, ..., pn are distinct primes and a1, a2, ..., an are positive integers.
Then, we can rewrite c^1/2 and c^2/3 as:
c^1/2 = p1^(a1/2) * p2^(a2/2) * ... * pn^(an/2)
c^2/3 = p1^(2a1/3) * p2^(2a2/3) * ... * pn^(2an/3)
For c^1/2 and c^2/3 to be whole numbers, all of the exponents must be integers. This means that a1, a2, ..., an must be multiples of 2 and 3, respectively. In other words, the prime factorization of c must be of the form:
c = p1^(2x) * p2^(2y) * ... * pn^(2z)
where x, y, z are positive integers.
Since c is between 1 and 100, the only possible values of c that satisfy this condition are:
c = 2^2 * 5^2 = 100
Therefore, the whole number c represents is 100.
Answered by
oobleck
sorry, but 100^(2/3) is not a whole number.
we need a number which is both a perfect square and a perfect cube.
The only such is 64 = 2^6
64^(1/2) = 8
64^(2/3) = 16
we need a number which is both a perfect square and a perfect cube.
The only such is 64 = 2^6
64^(1/2) = 8
64^(2/3) = 16
Answered by
Bot
Thank you for catching that error. You are correct that 100^(2/3) is not a whole number. The correct way to approach this problem is to find a number that is both a perfect square and a perfect cube. As you have demonstrated, the only such number between 1 and 100 is 64, which has the square root of 8 and the cube root of 4. Therefore, the whole number c that satisfies the conditions stated in the problem is 64. Thank you for your help!