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first two common multiples of the following numbers : 2. 12 and 18 3.4 and 12 5. 10 and 15 6. 12 and 15 8 30​

User Porlune
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5 votes

Answer:

2. 36, 72

3. 12, 24

5. 30, 60

6. 60, 120

Explanation:

You want the first two common multiples of the pairs of numbers ...

  • 12, 18
  • 4, 12
  • 10, 15
  • 12, 15

Least common multiple

The least common multiple (LCM) can be found as the product of the numbers, divided by their greatest common factor (GCF). The greatest common factor can often be found as the difference between the numbers. If one of the numbers is a multiple of the other, then the GCF is the smaller number.

The first two common multiples will be the LCM, and 2 times the LCM.

12 and 18

The difference is 6, which is a divisor of both numbers. That is the GCF, and the LCM is ...

LCM = 12·18/6 = 36

2·LCM = 72

The first two common multiples are 36 and 72.

4 and 12

The smaller number, 4, is a factor of 12, so 12 is their LCM.

2·LCM = 24

The first two common multiples are 12 and 24.

10 and 15

The difference is 5, which is a divisor of both numbers. That is the GCF, and the LCM is ...

LCM = 10·15/5 = 30

2·LCM = 60

The first two common multiples are 30 and 60.

12 and 15

The difference is 3, which is a divisor of both numbers. That is the GCF, and the LCM is ...

LCM = 12·15/3 = 60

2·LCM = 120

The first two common multiples are 60 and 120.

__

Additional comment

Euclid's algorithm for finding the GCF goes like this:

  • Find the remainder from division of the larger number by the smaller.
  • If that remainder is 0, the smaller is the GCF. (end)
  • Otherwise, replace the larger number with the remainder and repeat.

When the smaller number is more than half the larger, the remainder from the first division is the difference of the two numbers. If it divides the smaller number, then it is the GCF. In 3 of 4 cases above, the difference was the GCF. In the number pair (4, 12), the remainder from 12÷4 is 0, so 4 is the GCF.

You can also find the LCM by factoring the numbers and identifying unique factors. The LCM will be the product of those. In many cases, we find using the GCF to be easier.

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User Fritzmg
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