Problem:
How many 3-digit positive odd multiples of 3 do not include the digit 3 ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be a 3 -digit positive odd multiple of 3 that does not include the digit 3 . There are 8 possible values for , namely , and 9 , and 4 possible values for , namely , and 9. The possible values of can be put into three groups of the same size: , and . Recall that an integer is divisible by 3 if and only if the sum of its digits is divisible by 3 . Thus for every possible pair of digits , the choices for such that is divisible by 3 constitute one of those groups. Hence the answer is .
OR
There are odd 3 -digit multiples of 3 . Those including the digit 3 have the form , or . There are 30 of the first type, where the number is one of . There are 15 of the second type, where the number is one of , 99. There are 17 of the third type, where the number is one of . The numbers , and 933 are each counted twice, and 333 is counted 3 times. By the InclusionExclusion Principle there are such numbers.
The problems on this page are the property of the MAA's American Mathematics Competitions