Using Euclid's algorithm, the first number you check is their difference:
60 - 48 = 12
Since 12 divides both numbers evenly, that is your GCF.
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If the difference does not divide both numbers evenly, you can repeat the procedure with the difference and the remainder from dividing the smallest of the other numbers by the difference. For example, ...
GCF(60, 44) = GCF(44 mod 16, 16) = GCF(12, 16) = 4
