Math Olympiad #29 Homework Solutions

1. The required multiples are 12, 15, 18, . . . , 225.  This can be rewritten as 4 x 3, 5 x 3, 6 x 3, . . . , 75 x 3.  So there are 72 multiples of 3 between 10 and 226.

2. The area of EFG is 9 square units.

3. 85 can be expressed as the product of 17 x 5 or 85 x 1.  In each of these ways, the larger factor represents the sum, and the smaller factor represents the difference.  Since the difference cannot be 1, 17 must be the sum.

4. The numbers which have 5 as a factor are: 5, 10, 15, 20, 25, 30.  The prime factorization of the product of these numbers will have seven 5s.

5. Pages 1 to 9 have 1 piece of type each (9 x 1 = 9).  Pages 10 to 99 have 2 pieces of type each (90 x 2 = 180).  Pages 100 to 150 have 3 pieces of type each (51 x 3 = 153).  So the total is 9 + 180 + 153 = 342 pieces of type.