101
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There are 70 composite numbers between 1 and 100 (inclusive). They are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 94, 95, 96, 98, 99, and 100.
The number 1 is neither prime nor composite, because prime numbers have exactly two whole factors, and composite numbers have more than two whole factors. There is only one whole factor for 1, and that is 1.
The other 29 numbers are prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 74, 79, 83, 87, 89, 91, 93, and 97.