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例1求72的所有正因数的乘积.分析与解因为72=2~3×3~2,故共有正因数(3+1)×(2+1)=12个:1,2,3,4,6,8,9,12,18,24,36,72,再将12个正因数配成(12/2=)6对:1与72,2与36,3与24,4与18,6与12,8与9,每对中,两数之积为72,由此可见,72的所有正因数的乘积是72~6.例2求36的所有正因数的乘积.分析与解因为36=2~2×3~2,故共有正因数:(2+1)×(2+1)=9个:1,2,3,4,6,9,12,18,36,将其中8个正约数配成(8/2=4)对:1与36,2与18,3与12,4与9,另外还有一个6.其中每对两数之积为36,由此可见,36的所有
Example 1 Find the product of all the positive factors of 72. Analysis and solution Because 72=2~3×3~2, there are total positive factors (3+1)×(2+1)=12: 1,2,3, 4,6,8,9,12,18,24,36,72, then 12 positive factors (12/2=) 6 pairs: 1 and 72, 2 and 36, 3 and 24, 4 and 18 , 6 and 12, 8 and 9, in each pair, the product of two numbers is 72, it can be seen that the product of all positive factors of 72 is 72~6. Example 2 Find the product of all positive factors of 36. Analysis and solution Because 36=2~2×3~2, there are total positive factors: (2+1)×(2+1)=9: 1,2,3,4,6,9,12,18,36. Among them, 8 positive divisors are paired (8/2=4) pairs: 1 and 36, 2 and 18, 3 and 12, 4 and 9, and there is another 6. The product of two pairs of each pair is 36. This is visible, all of 36