How many positive integers that are multiples of 3 are also divisors

Question

How many positive integers that are multiples of 3 are also divisors of 42 ?

A. One
B. Two
C. Three
D. Four
E. Five

Kiara229 · Answer

IMO D

I solved the problem by doing the prime factorization of 42 - 2,3,7 and used all combinations of products with the integer 3. Therefore, I came to 4 multiples of 3 that can divide 42 evenly. (3,2*3,3*7,2*3*7)

Aravind04 · Answer

Please correct me if I am wrong:

divisors of 42 are 1,2,3,6,7,14,21,42 total 8 Divisors

out of these 8 Divisors
 3,6,21& 42 are multiples of 3 
hence 4 numbers Option D

Sameer1 · Answer

Hey , can you elaborate this method a bit?

Thanks.

egmat · Answer

Hi Sameer1,

The method Kiara229 used is a slick shortcut, so let me unpack the logic behind it.

Start with the prime factorization:  42 = 2 × 3 × 7 .

The key idea:  a divisor of  42  that is a  multiple of 3  must contain that  3  as one of its building blocks. If it didn't have a  3  in it, it couldn't be a multiple of  3 . So lock the  3  in place first.

Now the only freedom you have is what to attach to that  3 , using the  leftover  primes  {2, 7} . For each leftover prime you make one simple yes/no choice - include it or not:

- 3 alone →  3 
- 3 × 2 →  6 
- 3 × 7 →  21 
- 3 × 2 × 7 →  42

That's  2  choices for the  2  (in or out) ×  2  choices for the  7  (in or out) =  4  divisors. Every one of them divides  42  and is a multiple of  3 , which gives you answer  D .

This is exactly the same list Aravind04 got by writing out all  8  divisors and picking the ones divisible by  3  - the prime-factor method just skips straight to them.

Quick one to lock it in:  how many divisors of  30  are multiples of  3 ?

-  30 = 2 × 3 × 5 . Fix the  3 , then choose freely from the leftovers  {2, 5} :  2  ×  2  =  4  of them ( 3 ,  6 ,  15 ,  30 ).

Same move every time: nail down the required prime, then count the on/off combinations of what's left.

Answer: D

How many positive integers that are multiples of 3 are also divisors

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How many positive integers that are multiples of 3 are also divisors

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