As a first step you need to find the factors of the numbers
25 = 5*5
63 = 9*7
The lowest common multiple must include factors so that each of the original numbers can be created out if it. In this case we get 5*5*9*7 =1575
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32 = 2*2*2*2*2
15 = 5*3
So LCM is 2*2*2*2*2*5*3 = 480
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12 = 2*2*3 = 2*(2*3)
30 = 2*3*5 = 5*(2*3)
In this case, both numbers are multiples of 6, so we only need to include 6 once.
LCM 2*5*(2*3) = 60
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I hope this is not too confusing
You don't actually need to factorize the numbers. All you need to do is find the GCD (greatest common divisor) of the numbers using Euclid's Algorithm:
63=2*25+13
25=1*13+12
13=12+1
12=12*1+0
so the GCD of 25 and 63 is 1.
Therefore the LCM is 25*63/1=1575
32=2*15+2
15=7*2+1
2=2*1+0
so the GCD of 32 and 15 is 1.
Therefore the LCM is 32*15/1=480
30=2*12+6
12=2*6+0
so the GCD of 30 and 12 is 6.
Therefore the LCM is 30*12/6=60