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what are the first three common multiples of 6,9,10

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3 Answers

find the LCM of 6, 9, and 10
 
first write each number as a product of prime factors
 
6=2x3
9=3x3
10=2x5
 
to be a multiple of 6, the LCM must contain all of the  prime factors of 6
LCM=2x3 so far
to be a multiple of 9, the LCM must contain all of the prime factors of 9, but we don't use those prime factors that we already have(we want the least common multiple)
we need two 3's but we already have one 3 so we take only one 3 from 9
LCM=2x3x3 so far
to be a multiple of 10, the LCM must contain all of the prime factors of 10, but we don't use those prime factors that we already have
we need a 2 and a 5 but we already have the 2 so we take the 5 only from 10
LCM=2x3x3x5=90
the next common multiples will be multiples of 90: 180, 270, 360...
 
 
Find out the first common multiple, then the second and the third multiple will be double and triple the first multiple.
 
To find the first common multiple:
 
6- 1x2x3
9- 1x3x3
10-1x2x5
 
Now multiply all different numbers and each addition 3 only.
1x2x3x3x5=90
 
So the answer is 90,180 and 270
Common factors, or common multiples? 6,9,and 10 do not share any common factors among the three numbers.
 
Assuming you're looking for common multiples, in short, we're looking a number that ends in zero (multiple of 10), is even AND the digits add to a multiple of 3 (rule for multiples of 6), and whose digits sum up to 9 (rule for all multiples of 9)
 
Thus, the common multiples would be 90, 180, 270, 360, 450, 540, 630, 720, 810, 900, 1080, 1170, 1260, ... This is a pretty cool pattern :)