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Q: Why is there no greatest common multiple for any pair of numbers?

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There is no "greatest common multiple" for any pair of numbers. If you meant greatest common factor, the answer is 3.

There is no greatest common multiple of these numbers... or indeed of any pair of numbers. Once you get a common multiple, you can multiply that by any integer, as large as you want, to get a larger common multiple.

Only if the magnitudes of two numbers are the same.

7 and 35

15 and 90 30 and 45

12 and 30 6 and 60

4 and 24 8 and 12

No.

6 and 9 are one pair of numbers with a common multiple of 18. Another pair are 9 and 18.

36 and 48 12 and any multiple of 12

Since 70 is a multiple of 14, it is the least common multiple. Or, you can determine it as you would any pair of numbers. The least common multiple of two numbers is the product of the two numbers divided by their greatest common factor. The greatest common factor of 14 and 70 is 14. Therefore, the least common multiple 14 x 70 ÷ 14 = 70.

There is no greatest common multiple of this or any pair of numbers. The least common multiple is 100 which is 5 times 20 and 2 time 50. However, 200,300,400, 500, 600, ..... are all common multiples as well.

Yes, the greatest common factor of a pair of numbers can equal one of the numbers if one of the numbers is a factor of the other.

7 and 35

A prime number

Any pair of prime numbers

48 and 36

No.

12,36

1

That happens when one of the numbers is a factor of the other.

12 and 24 is one such pair.

Two pairs of numbers with a GCF of 8 and an LCM of 360 are 360 and 8 and 72 and 40

when one of the numbers is a multiple of the other, or when the other number is one.