**Answer is: 49 and 77**

**METHOD 1:**

*Strategy: Examine the possible prime factors of N.*N is composite, so it is the product of two or more (not necessarily different) primes. None of the prime factors can be 2, 3, or 5, for then at least one of the fractions given would not be in lowest terms. The possible prime factors of N are 7, 11, 13, 17, etc. The composites using these factors are 7 × 7, 7 × 11, 7 × 13, 11 × 11, etc. Because only the first two composites are between 20 and 80,

**the only possible values of N are 49 and 77**.

**METHOD 2:**

*Strategy: Determine the whole numbers between 20 and 80 that don't work.*Start with the numbers 21, 22, 23, . . ., 79. 2/N , 3/N , and 5/N are in lowest terms, so eliminate multiples of 2, 3, and 5. This leaves 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, and 79. N is composite, so eliminate primes, leaving only 49 and 77 The possible values of N are 49 and 77.