Factorisation is the quickest way to solve it. Since after the factorisation of 10, we'll already have the prime factors of 10, we just need to factorize 21 to get the count for 210.

A doubt regarding this question

why we can't take -1 or -10 as a factor

as per the article -Technically, in the world of mathematics - the answer to this question is yes. There are negative factors and multiples of numbers. For example, -4 and -3 are factors of 12, -24 is a multiple of 3, and the factors of 4 would be -4, -2, -1, 1, 2, 4.

But what exactly does this mean in relation to the GRE and/or the GMAT?

For the GRE: This concept is mentioned in the ETS GRE Math Conventions PDF. You can find it here. The factors of an integer include both the positive and negative integers. This also applies for multiples.

Therefore, technically, if a GRE question were to ask "How many factors does 6 have?" The answer would be 8: (-6, -3, -2, -1, 1, 2, 3, 6). With that said, questions are very likely to specify "positive factors" or "positive integers" on the GRE. We haven't seen an official question before that required knowing that negative factors/multiples exist - but that doesn't mean those questions won't show up at all so you still want to be diligent about this.

we should consider them also.

P.S-why there is a restriction
never encountered this issue on GC

Prime factors of 10- 1x10, 2x5
Prime factors of 210- 3x5x2x7

Hence Proved
4=4

therefore C

Please give a kudos

I think distinct positive factor is the correct question
Total no Distinct factors of 10 are 8 then considering all the negative factors

So if we consider this should.t the answer to this question be A, since factors of 10 are 1,2,5,10,-1,-2,-5,-10 and prime factors for 210 are 2,5,7,3

Quantity A

The number of distinct factors of 10

1, 2, 5, 10 = 4

or including negative factors

1, -1, 2, -2, 5, -5, 10, -10 = 8

Quantity B

The number of distinct prime factors of 210

2, 3, 5, 7 = 4

or including negative factors

2, -2, 3, -3, 5, -5, 7, -7 = 8

Either way, BOTH quantities are EQUAL

The answer is C.

You are implying negative primes. Can negative numbers be prime though?

Absolutely NOT. primes cannot be negative

We need to compare:
- Quantity A: The number of distinct factors of 10.
- Quantity B: The number of distinct prime factors of 210.

Step 1: Find the Number of Distinct Factors of 10 (Quantity A)
1. Prime Factorization of 10:

$$
\(10=2 \times 5\)
$$

2. Formula for Number of Factors:

If $\(n=p_1^a \times p_2^b \times \cdots \times p_k^z\)$, then the number of distinct factors is:

$$
\((a+1)(b+1) \ldots(z+1)\)
$$

3. Apply to 10:

$$
\(10=2^1 \times 5^1 \Longrightarrow \text { Number of factors }=(1+1)(1+1)=4\)
$$

4. List of Factors (Verification):

$$
\(1,2,5,10 \quad \text { (Total: } 4)\)
$$


Step 2: Find the Number of Distinct Prime Factors of 210 (Quantity B)
1. Prime Factorization of 210:

$$
\(210=2 \times 3 \times 5 \times 7\)
$$

2. Count of Distinct Prime Factors:

The distinct prime factors are 2, 3, 5, 7, totaling 4.
Step 3: Compare Quantities A and B
- Quantity A: 4 (distinct factors of 10 )
- Quantity B: 4 (distinct prime factors of 210 )

Conclusion
Quantity $\(\mathrm{A}=\)$ Quantity B

The question doesn't mention that we need distinct positive factors of 10, though. It can have distinct negative factors too. Editing the question will be helpful.

Prime factorization involves breaking down a number into its prime factors. While negative numbers can be factored, they are not considered prime themselves. A prime number is defined as a positive integer greater than 1 with only two factors: 1 and itself.

Therefore when you perform prime factorization, you break down a composite number into its prime numbers and not but still positive

I suggest to read more here https://gre.myprepclub.com/forum/gre-ma ... tml#p95905 and here https://gre.myprepclub.com/forum/gre-qu ... tml#p51913

In the case of the question above for QA we do have four factors which are 1,10,2 , and 5

And in QB we do have four primes 2,3,5,and 7

C is the answer

IMO, the answer cannot be C unless question specifically mentions that Quantity A is "The number of distinct POSITIVE factors of 10". Since it doesn't mention the word POSITIVE, Quantity A can include negative factors too, making it a total of 8 factors. The Quantity B remains 4 though, restricted by the "distinct prime factors".