find two pairs of numbers with the given number as their least common multiple the numbers are 10 36 60 105
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To find two pairs of numbers with a given number as their least common multiple (LCM), we can use the prime factorization method.
Let's start by finding the prime factorizations of the given numbers:
10 = 2^1 * 5^1
36 = 2^2 * 3^2
60 = 2^2 * 3^1 * 5^1
105 = 3^1 * 5^1 * 7^1
To find the LCM, we need to take into account the highest power of each prime factor present in the given numbers.
Now, let's look at each prime factor individually and choose the highest power of each that appears in any of the given numbers:
For 2: the highest power is 2^2 = 4
For 3: the highest power is 3^2 = 9
For 5: the highest power is 5^1 = 5
For 7: the highest power is 7^1 = 7
Multiplying these prime factors together, we obtain the LCM:
LCM = 2^2 * 3^2 * 5 * 7 = 4 * 9 * 5 * 7 = 1260
To find two pairs of numbers with 1260 as their LCM, we need to factorize it again:
1260 = 2^2 * 3^2 * 5 * 7
Now, we can choose any two numbers that have a subset of these prime factors.
For example, one possible pair would be:
1. 2^2 * 3^1 * 5^1 * 7^1 = 420
2. 2^1 * 3^2 * 5^1 * 7^1 = 630
Another possible pair would be:
1. 2^2 * 3^1 * 5^1 * 7^0 = 60
2. 2^0 * 3^2 * 5^1 * 7^1 = 315
So, two pairs of numbers with 1260 as their LCM are (420,630) and (60,315).