A certain number has a total of 16 factors. The square of this number

Question

A certain positive integer has a total of 16 factors. The square of this number cannot have which one of the following total factors?

(A) 31

(B) 45

(C) 49

(D) 63

(E) 87

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gmatophobia · Answer

If a number is represented in the form  p^x * q^y * r^z ..  where p,q,r and x, y , z are positive integers

Total number of factors = (x+1)(y+1)(z+1)

Is given that x consists of total 16 factor.

16 =  2^4 , thus the total number of factors can be formed by multiplying the 2's in any order.

Lets assume that the number is N

For ex-
16 could have been formed by multiplying 4 * 4, or 2 * 8 or may be just 16* 1.

Option analysis

A) 31

N =  p^{15}

Total factors of N will be 16.

N ^ 2   =  p^{(2*15)}

Total factors of  N^2  will be 31.

Hence this is possible

B) 45

N =  p^1 * q^7

Total factors N will be 2 * 8 = 16.

N ^ 2   = p^2 * q^{14}

Total factors of  N^{2}  will be 3 * 15 = 45.

Hence this is possible

C) 49

N =  p^3 * q^3

Total factors N will be 4 * 4 = 16.

N ^ 2   = p^6 * q^{6}

Total factors of  N^{2}  will be 7 *7 = 49.

Hence this is possible

D) 63

N =  p^1 * q^1 * r^3

Total factors N will be 2 * 2 * 4 = 16.

N ^ 2   = p^2 * q^2 * r^6

Total factors of  N^{2}  will be 3 * 3 * 7= 63.

Hence this is possible

D) 87
 Only remaining option 
 
IMO - E

ankitksingh · Answer

@Bunuel Can you please help with the OE?

Bunuel · Answer

A certain positive integer has a total of 16 factors. The square of this number cannot have which one of the following total factors?

(A) 31
(B) 45
(C) 49
(D) 63
(E) 87

Finding the Number of Factors of an Integer 
 
First, make the prime factorization of an integer  n = a^p * b^q * c^r , where  a ,  b , and  c  are prime factors of  n , and  p ,  q , and  r  are their respective powers.

The number of factors of  n  will be expressed by the formula  (p+1)(q+1)(r+1) .  NOTE:  this will include 1 and  n  itself.

Example:  Finding the number of all factors of 450:  450 = 2^1 * 3^2 * 5^2

Given the number has 16 factors. So (p + 1)(q + 1)(r + 1)... = 16.

All valid factorizations of 16 are:
16
8 * 2
4 * 4
4 * 2 * 2
2 * 2 * 2 * 2

Check each case.

Case 1: 16. 
The number is of the form prime^15. 
It's square would be p^30, which would have (30 + 1) = 31 positive factors.

Case 2: 8 * 2 
The number is of the form p^7 * q^1, since (7 + 1)(1 + 1) = 16.
Its square would be p^14 * q^2, which would have (14 + 1)(2 + 1) = 45 positive factors.

Case 3: 4 * 4 
The number is of the form p^3 * q^3, since (3 + 1)(3 + 1) = 16.
Its square would be p^6 * q^6, which would have (6 + 1)(6 + 1) = 49 positive factors.

Case 4: 4 * 2 * 2 
The number is of the form p^3 * q^1 * r^1, since (3 + 1)(1 + 1)(1 + 1) = 16.
Its square would be p^6 * q^2 * r^2, which would have (6 + 1)(2 + 1)(2 + 1) = 63 positive factors.

Case 5: 2 * 2 * 2 * 2 
The number is of the form p^1 * q^1 * r^1 * s^1, since (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 16.
Its square would be p^2 * q^2 * r^2 * s^2, which would have (2 + 1)(2 + 1)(2 + 1)(2 + 1) = 81 positive factors.

Thus, the square can have 31, 45, 49, 63, or 81 positive factors, so 87 is not possible.

Answer: E

A certain number has a total of 16 factors. The square of this number

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A certain number has a total of 16 factors. The square of this number

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