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three pairs of numbers for which the least common multiple equals the product f the two numbers

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2 Answers

Juliana,
 
For two numbers (lets call them a & b), the least common multiple is the smallest number that is an integer multiple of both a and b.  For example, given the numbers 12 and 18, their least common multiple is 36, because 3x12=36 and 2x18=36.  A least common multiple can never be greater than the product of the two numbers because b times a is the b-th multiple of a, and the a-th multiple of b.  For example, the least common multiple of 7 and 11 is 77 because multiples of 7 are 14, 21, 28, 35, 42, 49, 56, 63, 70 and 77 and multiples of 11 are 22, 33, 44, 55, 66, 77, so 77 is the first one that is a multiple of both numbers.  (BTW, 7 & 11 can be your first pair).
 
The least common multiple of two numbers is the product of the two numbers when those numbers are relatively prime to each other, that is to say that the two numbers do not share a factor in common (except for 1).  For example, the least common multiple of 9 and 8 is their product 72.  While neither 9 nor 8 are prime numbers, the factors of 9 are 3*3 and of 8 are 2*2*2.  Since they share no factors in common, their product will be the least common multiple.  (Multiples of 9 are 18, 27, 36, 45, 54, 63, and 72; Multiples of 8 are 16, 24, 32, 40, 48, 56, 64, and 72).  (BTW, 8 & 9 could be your 2nd pair.
 
So to construct a pair of numbers whose product is their least common multiple all you need to do is create two lists of prime factors, one for each number, where none of their factors overlap.  While a number can be on one list multiple times (like the number 2 for 8 or the number 3 for 9), the same number cannot be on the list for both a and b.  The first six prime numbers are 2, 3, 5, 7, 11, and 13.  You can mix and match them however you'd like (following the rule above) to create your third pair of numbers.  If you comment your choice, I'd be happy to confirm whether you have successfully constructed such a pair.
 
I hope this helps.  John
When the LCM of two numbers equals the product of the two numbers, the two numbers have no common factors other than the number one.
Examples: 4 and 9, LCM=4*9=36
                  8 and 15, LCM=8*15=120
                 12 and 25, LCM=12*25=300
Counter example: 12 and 15, LCM=60 not 12*15 which equals 180 because 12 and 15 have a common factor of 3(the GCF) and if you divide 180 by 3 you get 60 !