In constructing the Sieve of Eratosthenes for 2 through 100, it was said that any composite in that range had to be a multiple of some prime less than or equal to 7 (since the next prime, 11, is greater than the square root of 100). Explain.
Since the next prime after 7 is 11, the smallest composite number whose prime factors are all greater than 7, is
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11 times 11 equals 121.11•11=121.
11 plus 11 equals 22.11+11=22.
1 times 11 equals 11.1•11=11.
Since this is
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less
greater
than 100, there are no
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prime
composite
numbers, from 2 to 100, whose prime factors are all greater than 7. Therefore,
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every
at least one
composite number from 2 to 100 must be a multiple of some prime number less than or equal to 7.