1) There are many numbers that divide 109 with a remainder of 4. List all twodigit numbers that have that property.
2) If a number ends in zeros, the zeros are called terminal zeros. For example, 520,000 has four terminal zeros, but 502,000 has just three terminal zeros. Let N equal the product of all natural numbers from 1 through 20: N = 1 x 2 x 3 x 4 x . . . x 20. How many terminal zeros will N have when it is written in standard form? 3) The XYZ club collected a total of $1.21 from its members with each member contributing the same amount. If each member paid for his or her share with 3 coins, how many nickels were contributed? Solutions 1) If 4 is subtracted from 109, the result is 105. Then each of the twodigit numbers that will divide 109 with a remainder of 4 will divide 105 exactly. Thus, the problem is equivalent to finding all twodigit divisors of 105. Since the prime factors of 105 are 3, 5, and 7, the divisors are 3 x 5, 3 x 7, and 5 x 7, or 15, 21, and 35. 2) Multiplication by 10 produces an additional terminal zero when a product is written in standard form. If 1 x 2 x 3 x . . . x 20 is written as a product of prime factors, it will contain four 5s and more than four 2s among many factors. Then part of the product can be written as: (5 x 2) (5 x 2) (5 x 2) (5 x 2) which can also be represented as 10 x 10 x 10 x 10. Therefore there are four terminal zeros in the product. 3) Since 121 = 11 x 11, the club has 11 members and each contributed $0.11. Each $0.11 share was paid in 3 coins which had to be 2 nickels and 1 penny. Then the 11 members contributed a total of 22 nickels.
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