No 5's Allowed!
Number Theory
Level
2
How many positive integers less than 1000 are odd but not a multiple of 5?
The answer is 400.
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1 solution
A number is odd if and only if the last digit is 1, 3, 5, 7 or 9.
An odd number is a multiple of 5 if and only if the last digit is 5.
Hence, an integer is odd but not a multiple of 5 if and only if the last digit is 1, 3, 7 or 9.
The other digits can be any value from 0 to 9, for a total of 10. Hence, there are 4 × 1 0 × 1 0 = 4 0 0 possibilities.