If n is a positive integer and the greatest common divisor of n and 16

Question

If n is a positive integer and the greatest common divisor of n and 16 is 4, and the greatest common divisor of n and 15 is 3, which of the following is the greatest common divisor of n and 150?

A. 3
B. 6
C. 12
D. 15
E. 30

Bunuel · Accepted Answer

The greatest common divisor of n and 16 is 4 --> n is a multiple of 4 but not a multiple of 8.
The greatest common divisor of n and 15 is 3 --> n is a multiple of 3 but not a multiple of 5. Eliminate D and E.

150=2*3*5^2 is not divisible by 12, thus the greatest common divisor of n and 150 cannot be 12. Eliminate C.
Both n and 150 are divisible by 6, thus their greatest common divisor cannot be 3. Eliminate A.

Answer: B.

sumitkrocks · Answer

Note GCD is the product of least power of common factors

GCD of n and 16 is n = n has a factor  2^2  
GCD of n and 15 is 3 = n has factor 3 and not 5

150 =  2^1*3*5^2  
GCD of 150 and n will be = 2*3 (We know n has factor which is multiple of 5 and minimum power of 2 &3 in 150 is 1 )
B is the answer IMO

kantapong · Answer

GCD(n, 16) = 4 => 4 is the factor of n
GCD(n, 15) = 3 => 3 is the factor of n
Hence, 12 is the factor of n (the power of 2 is no more than 2; the power of 3 is no more than 1 and power of 5 is none)

GCD(n, 150) = GCD(n, 5^2 x 2 x 3) = 6 (Common factor of 2 is just 1 and common factor of 3 is 1)
Ans: B

ueh55406 · Answer

GCD is common prime in lowest powers.

greatest common divisor of n and 16 is 4

16=2^4 
 n=2^2 * Something

the greatest common divisor of n and 15 is 3

15=3*5 
 n=3* something .

So from the above two, we know  n= 2^2*3*something 
 and  150= 15*10= 3*5*5*2

GCD is common prime in lowest powers.

so, the common primes between n and 150 = 2 and 3 =   2*3 = 6

B is the right Answer.

asishron29181 · Answer

HCF of (n,16) = 4
n is a multiple of 4.

HCF of (n,15) =3
n is a multiple of 3

n is multiple of both 3 and 4
n is a multiple of 12
Let n =12x

HCF of (150, 12x) 
Prime factorize 150 = 5^2 * 2* 3
12x = 2^2 * 3 * x

HCF = 2 * 3 = 6

IMO B

HarshaBujji · Answer

If GCD(n,16) = 4;
n is of 4k for where k is not divisible by 2; =>4(2k+1).

If GCD(n,15) = 3;
Which  mean n is not divisible by 5 , and is in 3m form

So GCD(n,150)=GCD( n,2*3*5^2 )=So it is 6.

IMO B

purplex · Answer

1. Possible values = 4, 12
2. Possible values = 3,6,9,12

Common = 12

GCD of 12 and 150 = 6

If n is a positive integer and the greatest common divisor of n and 16

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If n is a positive integer and the greatest common divisor of n and 16

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