How many positive integers less than 20 can be expressed as

The numbers must be of the form \(2a+3b,\) where \(a\) and \(b\) are positive integers.
The smallest number is \(5 = 2*1 + 3*1.\) Starting with \(5\), we can get all the other numbers by adding either \(2\) or \(3\) to the already existing numbers on our list. Adding either \(2\) or \(3\) to \(2a+3b\) will give another number of the same form.
So, after \(5\), we get \(5+2=7, \,5+3=8, \,7+2=9, \,8+2=10,...\) We will get all the numbers up to \(19\) inclusive, except \(1,2,3,4,\)and \(6,\) because once we have \(7\) and \(8,\) by adding \(2\) all the time we can get any odd or even number.
We get a total of \(19-5=14\) numbers.

Answer A

Note: In fact, any integer \(n\) greater than 6 has at least one representation of the form \(2a+3b.\) If \(n\) is odd, then \(n-3>2\), so we can take \(b=1\) and \(a=\frac{n-3}{2}.\) If \(n\) is even, being greater than \(6\), \(n-6\) is a positive multiple of \(2\). Now we can take \(b=2\) and \(a=\frac{n-6}{2}.\)
If the question would have been the same but for integers less than \(100\), then the answer would be quite easy, \(99 - 5 = 94.\)