Question

2. Observe that 49 = 4 × 9 + 4 + 9.

(a) Find all other two-digit numbers which are equal to the product of their digits plus the sum of their digits.

(b) Are there three-digit numbers which are equal to the product of their digits plus the sum of their digits? Prove your answer.

Answer 1

(a) Let our two-digit number be $10a+b$. Then

$$10a+b=a\times b+a+b$$

$$a\times b=9a$$

Since $a\not=0,$ we have $b=9.$

Two-digit numbers which are equal to the product of their digits plus the sum of their digits: $19, 29, 39, 49, 59, 69, 79, 89, 99.$

(b) Let our three-digit number be $100a+10b+c$. Then

$$100a+10b+c=a\times b\times c+a+b+c$$

$$a\times b\times c=99a+9b$$

Since $a\not=0,$ we have $b\not=0, c\not=0.$

$$b\times c\leq9\times9=81<99$$

Then

$$a\times b\times c<99a+9b.$$

Therefore there is no three-digit number which is equal to the product of its digits plus the sum of its digits.