There is no such number.
Since if x were the largest multiple of 3 and 7 then what about 2x?
2x would be a multiple of 3 since x is a multiple of 3;
2x would be a multiple of 7 since x is a multiple of 7; and
2x is bigger than x. So x cannot be the largest.
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The largest multiple of 3 and 7 can be found by finding the least common multiple (LCM) of the two numbers. The LCM of 3 and 7 is 21. Therefore, the largest multiple of 3 and 7 is 21.
There is no largest common multiple of any two numbers: whatever common multiple is claimed to be the largest can be increased to an even larger common multiple by adding the lowest common multiple for the numbers (which for 2 and 3 is 6).
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The factors of 21 are 1, 3, 7, 21. The factors which are not a multiple of 7 are 1 and 3.
The (LCM) Least Common Multiple of 7 and 3 is 21 because that is the LEAST number that both 3 and 7 go into.