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The previous activities illustrate the difference between a necessary and a sufficient condition. We have shown that every prime greater than 3 is located in either the fourth or sixth columns of our grid. This means that it is necessary for the number to be located in one of those two columns if it is prime and greater than 3.
However, we also found numbers in those columns that are composites (not prime); thus, that location is not a sufficient condition for a number to be prime. This type of thinking is very useful when analyzing relationships in mathematics.
To give another example: For a number to be divisible by 6, it is necessary, but not sufficient, that it is divisible by 3. Conversely, for a number to be divisible by 3, it is sufficient, but not necessary, that it is divisible by 6.
It is necessary for the rest of the primes to fall in either the fourth or sixth columns, but that is not a sufficient condition.
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