Is the number of factors of A equal to 36?

Question

(1) A = B x C; B and C have exactly one common factor.

(2) B and C each have exactly Five factors; A = B x C.

uc26 · Answer

We need the powers of the unique prime factors of A to be able to answer the question.

1) We do not have info on the total factors of B and C. Not sufficient.
2) We do not have info on the common factors between B and C. Not Sufficient.
1&2) We have info on the total factors of B and C and the common factors. Sufficient to answer the question.

Answer: C

Archit3110 · Answer

Is the number of factors of A equal to 36?

1) A = B x C; B and C have exactly one common factor.

2) B and C each have exactly Five factors; A = B x C.

target total factors of A= 36 ; 
#1
A = B x C; B and C have exactly one common factor.
value of B*C can be of any integer value eg 10*2 ; 10*4 or 2*1 ; 3*1 factors of A will vary
insufficient
#2
B and C each have exactly Five factors; A = B x C.
B & C must be square of prime number eg A= 2^4*3^4 or 5^4*7^4
so total factors A = 25 ; sufficient
OPTION B

ashwini2k6jha · Answer

Answer D
State 1:sufficient
A=BXC,
both have one common factor means,both are prime number
hence number of factor for A=2*2=4
State:2 sufficient
Number of factors for A= 5x5=25

techwave · Answer

Is the number of factors of A equal to 36?

1) A = B x C; B and C have exactly one common factor.

2) B and C each have exactly Five factors; A = B x C.

Ans: E

1) B and C exactly one common factors . THis means lots of possibilites so not sufficient.

2) B and C each have exactly five factors.

if a number has exactly five factors then it must be of the form p^4, where p is a prime number. so again lots of possibiites. Insufficeint.

Together not sufficeint aswell.

MayankSingh · Answer

IMO B

Is the number of factors of A equal to 36?

1) A = B x C; B and C have exactly one common factor.

Let B= 2 X 3 , C = 2 X 5
A= 2 X 3 X 2 X 5 = 2^2 X 3 X 5 
No. of factors = (2+1) X (1+1) X (1+1) = 12

But if , B = 2 X 3^3 & C = 2 X 11^2
A = 2^2 X 3^3 x 11^2
No. of factors = 3 X 4 X 3 =36

Not Sufficient

2) B and C each have exactly Five factors; A = B x C.

B = P ^ 4 , P = Prime No.
C= q ^ 4 , q= Prime no.
A= p ^ 4 x q ^ 4
Now if , p not equal to q
No of Factors = 5 x 5 = 25
And if p=q
No of factors = 9

In both cases not equal to 36 
Sufficient.

Kinshook · Answer

Asked: Is the number of factors of A equal to 36?

1) A = B x C; B and C have exactly one common factor. 
HCF(B,C)=1; B & C are co-prime
Since B & C are unknown
NOT SUFFICIENT

2) B and C each have exactly Five factors; A = B x C. 
Let  p_1  and  p_2  be prime numbers
 B = p_1^4; C=p_2^4; A=p_1^4*p_2^4 
If  p_1 = p_2 = p; A=p^8 ; A has 9 factors
But if  p_1 
eq p_2 ;  A=p_1^4*p_2^4 ; A has 5*5 = 25 factors
Number of factors of A is NOT equal to 36.
SUFFICIENT

IMO B

Deep99 · Answer

Statement 1-
B and C are coprimes (i.e. highest common factor (HCF) is 1)
And A is a product of both
Not sufficient

Statement 2-
No. Of factors of B= 5 
Consider B to be of the form p^(a) where p is a prime
No. Of factors of B = a+1
=> a = 4
No. Of factors of C= 5
Consider C to be of the form q^(b) where q is a prime
No. Of factors of C = b+1
=> b = 4
Since we don't know if B amd C are co-primes we can't comment on the number of factors of A
Not sufficient

Statement 1 and 2
If B and C are co primes
No. Of factors of A= (a+1)*(b+1)= 25

Hence sufficient
Ans: C

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tyson1990 · Answer

From statement (1), take B & C with 1 common factor.

Lets take, B = 2 . 3^x, C = 2 . 5^y

A = B X C = 2^2 . 3^x . 5^y
No of Factor (A) = (2+1).(x+1).(y+1)
This can be any multiple of 3.

Hence, St(1) is not sufficient.

From Statment (2), B & C have exactly 5 factor.

Case:1
B = 2^4, C = 5^4
A = 2^4 . 5^4
No of Factor (A) = (4+1).(4+1) = 25

Case:2
B = C = 2^4
A = 2^4 . 2^4 = 2^8
No of Factor (A) = (8+1) = 9

In both cases, No of Factor (A) can never be 36.

Hence, St(2) is sufficient.

urshi · Answer

Answer : C

Statement 1 : Just provides the information that the factors B & C are mutually exclusive except for 1. This doesn't not provide any information on the number of factors for A

Statement 2 :2) B and C each have exactly Five factors; A = B x C

This total number of factors of A will be 36 only if we know that the factors of B & C are mutually exclusive.

Hence, statement 2 is insufficient.

We need both Statement 1 & 2 to answer the question.

Hence , Answer - C

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Lipun · Answer

ST1: Not sufficient info.

ST2: Only numbers with exactly 5 factors are 4th power of prime number. This implies B=X^4 and C=Y^4.
So, A = X^4 * Y^4. Total factors of A are (4+1)*(4+1) = 25 < 36. Sufficient.

Ans: B

anupam87 · Answer

statement 1 is insufficient. 
Statement 2 - 
if P is prime number then P has two factors {1,P}, P^2 will have 3 factors - {1,P,P^2} and P^4 will have 5 factors {1,P,P^2,P^3,P^4} . 
So, B and C is of the form (Prime)^4. A will have - (4+1)(4+1) = 25 factors. 
Hence, B is sufficient to answer the question.

smmc29 · Answer

(1) A = BxC with B and C have only 1 common factor

-> Let us say x is their common factor. This means that x should be equal to 1. But the statement lacks information on how many unique prime factors B and C have. B can be b1^a x b2^y ..., and the same goes for C. Clearly insufficient.

(2) A = BxC, B and C have 5 factors each.

-> The only way for us to have an odd number of factors is when we have a perfect square. Therefore the statement shows that B and C are both perfect squares. For a perfect square we have a total of 3 factors (remember b^n has a total of n+1 factors). In order for us to have 5 number of factors, we need n to be equal to 4. If n is equal to 4, therefore we can express B in terms of b^4, and C in terms of c^4 where b, and c are unique prime factors of B and C respectively. If this is the case then we have a total of (4+1)(4+1) = 25 factors for A. 
-> However, if b=c, we can see that A = c^8 or b^8. In this case, we will have a total of (8+1) = 9 factors for A.

Either case, we can see that A cannot have 36 factors -> SUFFICIENT.

Another way to think of (2) is that we have perfect squares as factors of A. Since A is a product of two perfect squares therefore A must also be a perfect square. Since A is a perfect square, therefore A cannot have an even number of factors.

ravigupta2912 · Answer

In statemet 1, if C and B are co-primes, then there is no chance that the product of co-primes can have 36 factors?

How is that statement not sufficient?

If B = 3, C = 5, A = 15 (total factors of A = 4)

Can anyone point me to another example where B and C are co-primes and A would have 36 factors?

Also, I don't agree with some of the responses here which quote numbers for statement 1 having 2 common factors (1 and the common factor quoted).

Fdambro294 · Answer

For the number of factors to equal 36 (unique/distinct positive factors of A), we need a prime factorization that will follow one of the following forms:

N = (p)^5 * (q)^5

N = (p)^8 * (q)^3

Etc.

Statement 1:

A = (B) (C)

If B and C share only one common factor, assuming B and C are positive integers, the one common factor must be 1 because every integer is divisible by 1.

If the only common factor between B and C is 1, then we know that B and C are co-prime and not equal.

However, other then this fact, we do not have any other information indicating the break down of the prime factorization.

For instance, B and C can both be prime numbers and:

A = (2) (3) ————> only 4 distinct positive factors

Or

A = (2)^8 (3)^3 ———-> (8 + 1) (3 + 1) = 36 distinct positive factors

Statement 1 is not sufficient

Statement 2:

Rule: for a positive integer to have an odd number of distinct factors necessarily means that it is a perfect square

Case 1:

B = C = (2)^4

B and C each have exactly 5 factors.

A = (B) (C) = (2)^4 * (2)^4 = (2)^8

9 distinct positive factors

Case 2: B and C are distinct integers in which case

B = (p)^4

C = (q)^4

A = (p)^4 * (q)^4

Number of distinct positive factors = (4 + 1) (4 + 1) = 5 * 5 = 25

It is not possible for A to have 36 distinct factors.

B - statement 2 sufficient

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Is the number of factors of A equal to 36?

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