GMAT Club World Cup 2022 (DAY 9): If k is a positive integer and k has

We know that k has 25 positive factors.

Hence k can have one of the two forms (p and q are prime numbers) -

Form 1

k = \(p^ {24}\)

Form 2

k = \(p ^ 4 * q ^ 4\)

Asked:

Factors of 5 * 7 * k

Statement 1

Both 5k and 7k have 50 distinct positive factors

Case 1:

k = \(p^ {24}\)

if \(p\neq{5}\)

Number of factors = 2 * 25 = 50

Same analysis is also applicable for 7.

Therefore total factors of 35k = 7 * 5 * \(p^{24}\) = 2 * 2 * 25 = 100

Case 2:

k = \(p ^ 4 * q ^ 4\)

if \(p\neq{5}\) and \(q\neq{5}\)

Number of factors = 2 * 5 * 5 = 50

Same analysis is also applicable for 7.

Therefore total factors of 35k = 7 * 5 * \(p ^ 4 * q ^ 4\) = 2 * 2 * 5 * 5 = 100

Hence A is sufficient

Statement 2

363 = 121 * 3 = \(11^2\) * 3

k = \(11^2\) * 3 * some factor

We know that k has 25 factors, so form 1 is not possible here. Which leaves us with form 2.

Therefore k is of Form 2

k = \(11^4 * 3^4\)

Therefore number of factors of \(7 * 5 * 11^4 * 3^4\) = 2 * 2 * 5 * 5 = 100

Hence B is also sufficient

IMO D

Need to know: Formula for total factors of an integer in prime factorized form x^a * y^b * z^c = (a+1)*(b+1)*(c+1)

k is a positive integer and has 25 positive factors. Now if a positive integer has an ODD number of FACTORS => it is a PERFECT SQUARE
k is a perfect square and since number of factors = 25, k = x^24 OR k = x^4 * y^4 (x and y are prime numbers)

We need to determine the number of factors of 35k: Now 35k = 5*7*k: So if k contains a 5 and/or 7 in its prime factorization, then it can give us different answers but if not, then the number of positive factors of 35k will be 100 (2*2*25)
So basically, questions needs us to determine if k has a 5 and/or 7 in its prime factorization. Let us examine the statements:

(1) Both 5k and 7k have 50 distinct positive factors.

k has 25 positive factors and both 5k and 7k have 50 {(1+1)*(25) for both} => k does not include a 5 or a 7
That means we have a definite answer that 35k has 100 positive factors. SUFFICIENT

(2) k is a multiple of 363

363 = 3*11*11 and k/363 is an integer. Now we already know that for k to have 25 positive factors, it needs to be of either of these 2 forms:
k = x^24 OR k = x^4 * y^4: Since k/363 is an integer we know that k already consists of 3 and 11 and it cannot have more than 2 prime numbers in its prime factorization. So k = 3^4 * 11^4
Which means that k does not include a 5 or a 7
Therefore, we have a definite answer that 35k has 100 positive factors. SUFFICIENT

Answer - D