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Find all multiples of 9 less than 10000 whose only digits are 1 and/or 8?

Mar 29, 2020

#1
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All of the digits digits greater than 100 need AT LEAST a 0 or 9 with digits 1 and 8

Therefore, the multiples that work are:

18

81.

Mar 29, 2020
#2
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I got a total of 7:

18, 81, 1818, 1881 ,8118, 8181, 8811

Mar 29, 2020
#3
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Sorry, I missed 1188. So, a total of 8?

Guest Mar 29, 2020
#4
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I agree with guest because for a number to be divisible by 9, the sum of its digits must be divisible by 9.

All numbers smaller than 10,000 has either 1, 2, 3, or 4 digits.

The only 1-digit number that contains only 1s and 8s are 1 and 8; both obviously fail.

The only 2-digit numbers are 11, 18, 81, and 88, giving us 18 and 81.

If they have more than two digits, we may want to use the "divisble by 9" rule.

The three-digit numbers have either:

3 1s and no 8s:  1 + 1 + 1 = 3, not divisble by 9

2 1's and 1 8:     1 + 1 + 8 = 10, not divisible by 9

1 1 and 2 8's:     1 + 8 + 8 = 17, not divisible by 9

no 1s and 3 8s:  8 + 8 + 8  = 24, not divisible by 9

The four-digit number have either:

4 1s and no 8s:  1 + 1 + 1 + 1 = 4, not divisble by 9

3 1s and 1 8:      1 + 1 + 1 + 8 = 11, not divisible by 9

2 1s and 2 8s:    1 + 1 + 8 + 8 = 18, divisible by 9:  1188, 1818, 1881, 8118, 8181, 8811

1 1 and 3 8s:      1 + 8 + 8 + 8 = 25, not divisible by 9

no 1s and 4 8s:  8 + 8 + 8 + 8 =  32, not divisible by 9

Mar 29, 2020