How many distinct factors do 165 and 300 have in common?

Question

A. 2
B. 3
C. 4
D. 5
E. 8

sashiim20 · Accepted Answer

By Prime factorization of  165  and  300 , we get;

165 = 3 * 5 * 11

300 = 2^2 * 3 * 5^2

Common factors between  165  and  300  is  one    3  and  one    5   = 3^1*5^1

Number of factors is expressed by the formula  = (p+1)(q+1)  ------------- (where  p  and  q  are powers of the prime factors.)

Number of common factors between  165  and  300   = (1+1)(1+1) = 2*2 = 4

Answer (C)...

Luckisnoexcuse · Answer

Factors of 165 (5*3*11) =  1 , 3 , 5 ,11, 15 ,33,55,165

Factors of 300 (2*2*3*5*5) =  1 ,2, 3 ,4, 5 ,6,10,12, 15 ,20,25,30,60,75,100,150,200,300

Distinct common Factors = 1,3,5,15

C

pushpitkc · Answer

When prime-factorized,  165 = 3*5*11  and  300 = 2^3 * 3 * 5^2

The common factors of the numbers 165 and 300 are 3 and 5.
Since we have been asked to find the distinct common factors, 
they will be 1,3,5 and 3*5(15) which is  4(Option C)

RJ7X0DefiningMyX · Answer

Carefully looking at the question stem, what is really being asked is the no of distinct factors of the GCD of 165 and 300

Now, we can easily get the GCD (also known as HCF) of 165 and 300, i.e. 15, factorising which we get 3^1 * 5^1

Now, to get the total no. of factors, (1+1)*(1+1) = 4, and hence, answer choice (C)

QuantMadeEasy · Answer

HCF of 165 and 300 is 15. 
No of distinct factors of 15 are 4.

C is correct.

ScottTargetTestPrep · Answer

165 = 5 x 33 = 5 x 11 x 3

300 = 12 x 25 = 2^2 x 3 x 5^2

Thus, the factors in common are 1, 3, 5, 15. (Recall that 1 is a factor of every number, even though it doesn’t play a role in the prime factorization of a number since it is not prime.)

Answer: C

TestPrepUnlimited · Answer

165 = 5 * 33 = 5 * 3 * 11. Since 300 doesn't contain an 11, but is a multiple of 3 and 5, GCF(300,165)=15. A trick here is  \frac{165}{5} = 165 * \frac{2}{10} = \frac{330}{10} = 33 ).

Since the GCF is 15, any common factor of 165 and 300 will be a factor of 15. Therefore 1, 3, 5, 15 are the common factors. We have 4, so pick C.

However, if the GCF was bigger, let's say 96, then counting would be more exhausting and we might miss a factor. In this case, we can find the number of factors of 96 systematically this way:

Prime factorize  96 = 32*3 = 2^5 * 3 . When constructing a factor of 96, we can choose from 0-5 multiples of two in the product. For the factor of three, we can choose either 0 threes or 1 multiple of three. Picking 0 multiples of both two and three means we construct the factor "1". Therefore there are  (5+1)*(1+1) = 6*2 = 12  options, thus 96 has 12 factors in total. We can abbreviate this method with (a+1)(b+1)(c+1) ... where a, b, and c are the powers of each factor.

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How many distinct factors do 165 and 300 have in common?

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