Answer :
Multiples of 400: 400, 800, 1200, 1600
Multiples of 600: 600, 1200, 1800
So the LCM is 1200
Multiples of 600: 600, 1200, 1800
So the LCM is 1200
Step-by-step explanation:
the LCM (least common multiple) is the product (multiplication) of the longest product strings of the prime factors.
prime factors of a number start with 2 (the smallest prime number). we keep dividing the number (and then the results) by 2 until the division is not delivering a whole number anymore. then we switch to the next higher prime number. and so on, until the result is 1.
prime factors of 400
400 : 2 = 200
200 : 2 = 100
100 : 2 = 50
50 : 2 = 25
25 : 2 no
25 : 3 no
25 : 5 = 5
5 : 5 = 1
the prime factorization of 400 is therefore
2×2×2×2×5×5
prime factors of 600
600 : 2 = 300
300 : 2 = 150
150 : 2 = 75
75 : 2 no
75 : 3 = 25
25 : 3 no
25 : 5 = 5
5 : 5 = 1
the prime factorization of 600 is therefore
2×2×2×3×5×5
the LCM is now the product of the longest multiplication chains for the prime factors :
the longest chain for 2 is
2×2×2×2 in 400
for 3 is
3 in 600
for 5 is
5×5 in 400 and 600
the LCM = 2×2×2×2 × 3 × 5×5 = 1,200.
which is much smaller than the "instinctive" product of 400 and 600 :
400×600 = 240,000
which is of course also a multiple of 400 and 600, but not the smallest one (LCM).
