How many positive integers less than 20 are equal to the sum of a positive multiple of 3 and a positive multiple of 4?

GRE Quant Problem Solving
1

Hi, I came across this problem in GregMat’s 2 month plan [ Week 1 Day 6 - Mega Quant Practice ]

How many positive integers less than 20 are equal to
the sum of a positive multiple of 3 and a positive
multiple of 4 ?
(A) Two
(B) Five
(C) Seven
(D) Ten
(E) Nineteen

On the session, it was solved as -
Factors of 3 - 3, 6, 9, 12, 15, 18
Factors of 4 - 4, 8, 12, 16
And option selected was (D) 10.

However, since 0 is also a positive integer less than 20, and a multiple of 3 and 4 (0 is a multiple of every integer), can we also include 0 as factors of 3 and 4?

So now, the possible sums are -

3, 4, 7, 10, 11, 13, 14, 15, 16, 17, 18, 19?

But, I don’t see 12 as an option. Am I missing something here?

2

zero is neither positive nor negative

To know more about zero :

3

Oh right, I got you :slightly_smiling_face: Thanks!

4

Im not understanding how this question was solved in the vide.
The numbers listed were for 3: 4,8, 12, 16 and then for 6: 4, 8, 12, then 7, 11, 15, 19 then 10, 14 , 18 to come up with 10. I have no clue where this came from. Pls help.
This is found in Arithmetic and algebra session 3

5

I think

6 = 3 \times 2
7 = 3 \times 1 + 4 \times 1
10 = 3 \times 2 + 4 \times 1

So adding 4 to any of the three would be a valid sum of a multiple of 3 and 4.

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