Maths Olympiad Prep

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Combinatorics Difficulty 3.0 Junior Find the answer

How many 44-digit positive integers (that is, integers between 10001000 and 99999999, inclusive) having only even digits are divisible by 5?5?

Pick one

Solution

The units digit, for all numbers divisible by 5, must be either 00 or 55. However, since all digits are even, the units digit must be 00. The middle two digits can be 0, 2, 4, 6, or 8, but the thousands digit can only be 2, 4, 6, or 8 since it cannot be zero. There is one choice for the units digit, 5 choices for each of the middle 2 digits, and 4 choices for the thousands digit, so there is a total of 4551=(B) 1004\cdot5\cdot5\cdot1 = \boxed{\textbf{(B) } 100} \qquad numbers.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.