Question

The number of all 3-digit numbers abc (in base10) for which $(a\times b\times c)+(a\times b)+(b\times c)+(c\times a)+a+b+c=29$ is

• A. 6
• B. 10
• C. 14
• D. 18

Answer 1

The correct option is C

14

$(a\times b\times c)+(a\times b)+(b\times c)+(c\times a)+(a+b+c)=29$

$(1+a)(1+b)(1+c)=30$

$2\times 3\times 5\to (a,b,c)\Rightarrow (1,2,3)\Rightarrow 6$

$1\times 6\times 5\to (a,b,c)\Rightarrow (0,5,4)\Rightarrow 4$

$1\times 3\times 10\to (a,b,1)\Rightarrow (0,2,9)\Rightarrow 4$

______

14