AMATYC SML Training Session #1

The questions in the competitions come from a wide variety of areas of mathematics. Today’s problems are all examples of number theory – questions dealing with patterns that emerge in the integers. Several of the following questions are taken directly, or with minor modification, from old competitions, while others are original or from other sources, but of a similar style. See how well you can do at these, and then we’ll discuss the mathematics behind the problems.

1. How many 0’s are at the end of the expansion of 2009! ?

a. 401 b. 496 c. 500 d. 502 e. 511

2. How many 1 digit prime numbers divide 3³² + 5⁵⁵?

a. 0 b. 1 c. 2 d. 3 e. 4

3. When written as a decimal number 2009²⁰⁰⁹ has D digits and leading digit L. Find D+L

a. 6638 b. 6639 c. 6640 d. 6641 e. 6642

4. What is the least number of prime numbers (not necessarily different) that 3185 must be multiplied by so that the product is a perfect cube?

a. 1 b. 2 c. 3 d. 4 e. 5

5. If each letter in the equation represents a different digit, what is the value of T?

a. 3 b. 4 c. 5 d. 6 e. 7

6. The number 877530p765q6 is divisible by both 8 and 11, with p and q both digits between 0 and 9 (inclusive). The number is also divisible by:

a. 7 b. 12 c. 16 d. 18 e. Not Enough Info

7. If p>5 is a prime number, what is the largest integer which must be a factor of p⁴-1?

a. 120 b. 150 c. 180 d. 240 e. 400

8. ACME employees receive paychecks every other Friday. Something very unusual happened in 2008 – they received 3 paychecks in the month of February. If Y is the next year with 3 February paydays, then the unit’s digit of Y is

a. 0 b. 2 c. 4 d. 6 e. 8

9. Call a composite number circumfactorable if all of its positive integer factors greater than 1 can be placed around a circle so that any two adjacent factors have a common factor greater than 1. How many composite numbers less than 200 are not circumfactorable?

a. 50 b. 52 c. 54 d. 56 e. 58