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Unformatted text preview: a,b ] /b is an integer. Hence a = ab [ a,b ] · [ a,b ] b = k · [ a,b ] b and so k | a . By deﬁnition of greatest common divisor, it follows that k ≤ ( a,b ), i.e., (1) ab [ a,b ] ≤ ( a,b ) . Finally, we claim that p := ab/ ( a,b ) is a common multiple of a and b , i.e., a | p and b | p . From the deﬁnition of greatest common divisor, we know that ( a,b ) | b , whence b/ ( a,b ) is an integer. Then p = ab ( a,b ) = a · b ( a,b ) Date : April 10th, 2008. 1 shows that a | p . The claim that b | p is proved similarly. It follows from the deﬁnition of least common multiple that (2) ab ( a,b ) ≥ [ a,b ] Combining ( 1 ) and ( 2 ) we obtain ( a,b )[ a,b ] = ab, as desired. To compute [612 , 696], ﬁrst use the Euclidean algorithm to show that (612 , 696) = 12. Now [612 , 696] = 612 × 696 (612 , 696) = 612 × 696 12 = 51 × 696 = 35 , 496 . 2...
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