Question

The number of positive integer solutions of a+b+c=60, where a is a factor of b and c, is

Answer 1

145

Answer 2

Solution:

We are given:

$$a + b + c = 60,\quad \text{with } a|b \text{ and } a|c.$$

1. Express $b$ and $c$ as:

   $$b = ak,\quad c = am,\quad k, m \in \mathbb{Z}^+$$

2. Substitute in the equation:

   $$a + ak + am = a(1+k+m)=60.$$

   Let $d = 1+k+m$. Then,

   $$a\cdot d = 60.$$

3. For given $a$ (divisor of 60), $d = \frac{60}{a}$ must be integer. Also, $k+m=d-1$ with $k, m \ge 1$. Let:

   $$k' = k-1,\quad m' = m-1 \quad \Rightarrow \quad k'+m' = d-3.$$

   The number of positive solutions $(k, m)$ equals the number of nonnegative solutions $(k', m')$ which is:

   $$\binom{(d-3)+2-1}{2-1} = d-2.$$

   This is valid for $d\ge3$.

4. List all divisor pairs $(a, d)$ such that $ad=60$ and $d\ge3$:

   $$\begin{array}{c|c|c} a & d=\frac{60}{a} & \text{Count }(d-2) \\\hline 1 & 60 & 60-2 = 58 \\ 2 & 30 & 30-2 = 28 \\ 3 & 20 & 20-2 = 18 \\ 4 & 15 & 15-2 = 13 \\ 5 & 12 & 12-2 = 10 \\ 6 & 10 & 10-2 = 8 \\ 10 & 6 & 6-2 = 4 \\ 12 & 5 & 5-2 = 3 \\ 15 & 4 & 4-2 = 2 \\ 20 & 3 & 3-2 = 1 \\ \end{array}$$

5. Total solutions:

   $$58+28+18+13+10+8+4+3+2+1 = 145.$$