Question

The product of the digits in the number 126 is 1 · 2 · 6 = 12.

a) How many three digit numbers have this property that the product of the digits is 12?

b) How many three digit numbers have the property that the product of the digits is 16? c) What is the first product that you cannot express as a product of the digits of a three digit number?

Answer 1

a) Any three digit number is of form ABC

Now we want A * B * C = 12 = 1.2.6 = 1.2.2.3 = 1.2².3

The choices are {1.4.3, 1.2.6}

Each has 3! ways to arrange themselves and so 6 choices for each element in set above

In total there are 6+6 = 12 choices and so 12 numbers are there whose product is 12

b) 16 = 12.8

There are choices { 1.2.8}

This can arrane in 3! ways and so total 6 choices

Hence there are 6 three digit numbers with product 16

c) 1 is the first product that we cannot express as products of a three digit number

Because 1 = 1.1.1 = 111

This is all 1s repeated