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Least Common Multiples

You will likely need to deal with the concept of multiples on the SAT test, especially the least common multiple. A multiple is any integer that can be evenly divided by another integer. For example, 18 can be evenly divided by 9, 6 and 3, so 18 is a multiple of 9, 6 and 3. 

Some students think of multiples as the reverse of factors. For example, if 18 is a multiple of 9, 6 and 3, then 9, 6 and 3 are possible factors of 19 (e.g., 2 × 9 = 18, 3 × 6 = 18). 

The least common multiple is a special type of multiple. It is the least (smallest) multiple that two numbers share. Another way of thinking about it is it is the smallest number into which you could evenly divide both numbers. You find the least common multiple in a similar way to finding the greatest common factor, buy using prime factorization. First, you find the prime factors for each number, then you multiply each factor by the maximum number of times it appears for either number. It’s easier to understand if you see an example, so here is one. 

What is the least common multiple of 6 and 8? 

First, find the prime factors: 

6 = 2 × 3    and    8 = 2 × 2 × 2 

The prime number two appears a maximum of three times for 8, so use 2 × 2 × 2. The prime number 3 appears a maximum of once for 6, so use one 3. 

Multiply them all together: 

2 × 2 × 2 × 3 = 24 

Thus, 24 is the least common multiple of 6 and 8. 

Now let’s try a more complex example. 

What is the least common multiple of 15 and 28? 

First find the prime factors:

15 = 3 × 5    and    28 = 2 × 14 = 2 × 2 × 7 

The number 2 appears a maximum of twice (2 × 2) in 28, the number 7 appears once, the number 3 appears once and the number 5 appears once. So multiply them together: 

3 × 5×2 × 2 × 7 = 420

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