The angry multiple of a natural number is the smallest multiple of the number formed only by the digits 0 and 1. For example, the angry multiple of 2, as well as of 5, is 10; the angry multiple of 3 is 111, and the angry multiple of 110 is itself.

a) What is the angry multiple of 20?

b) What is the angry multiple of 9?

c) What is the angry multiple of 45?

d) What is the smallest natural number whose angry multiple is 1110?
Solution

a) The first multiples of 20 are 20, 40, 60, 80, and 100. Therefore, the irate multiple of 20 is 100.

b) If the digits of a number divisible by 9 are only 0 and 1, this number must contain at least nine digits 1. For this multiple to be the smallest possible, it must have the fewest number of digits; thus, the irate multiple of 9 is 111111111.

c) A multiple of 45 is a multiple of 5 and 9; therefore, its units digit is 0 or 5, and the sum of its digits is divisible by 9. Since irate multiples are formed only by the digits 0 and 1, it follows that the irate multiple of 45 must have 0 as its units digit; thus, this multiple is 1111111110.

d) The number 1110 is the smallest number that has only the digits 0 and 1 and is, at the same time, a multiple of 3 (since the sum of its digits is 3) and a multiple of 2 (since its last digit is 0). Therefore, 1110 is the irate multiple of 6. Since the irate multiples of 1, 2, 3, 4, and 5 are, respectively, 1, 10, 111, 100, and 10, it follows that the smallest number whose irate multiple is 1110 is 6.