To locate both the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of numbers, you usually start off the equal way: you locate the top factorizations of the 2 numbers.
Then (here is the trick!) you positioned the elements into a pleasant neat grid of rows and columns, evaluate and contrast, after which, from the table, take most effectively what you want.
Find the GCF and LCM of 2940 and 3150.
First, I want to component each of the numbers they have got given me:
Multiples and Least Common Multiples
I began out by dividing 2940 through the smallest top that could match into it, being 2. This left me with any other even wide variety, 1470, so I divided through 2 again. The result, 735, is divisible through 5, however, 3 divides in additionally, and it is smaller, so I divided through 3 to get 245. This isn’t always divisible through 3 however is divisible through 5, so I divided through 5 and got 49, which is divisible through 7.
Now I’ll follow the equal sequential-department procedure to 3150:
I’ve divided every of the given numbers through the smallest primes that match into them, till I ended up with a top result. The factorizations maybe examine from the numbers alongside the out doors of the sequential divisions. So my top factorizations are:
2940 = 2 × 2 × 3 × 5 × 7 × 7
3150 = 2 × 3 × 3 × 5 × 5 × 7
I will write those elements out, all pleasant and neat, with the elements coated up consistent with occurrence:
2940: 2×2×3 ×5 ×7×7 3150: 2 ×3×3×5×5×7
This orderly list, with every component having its very own column, will do the maximum of the paintings for me.
The Greatest Common Factor, the GCF, is the biggest (“greatest”) wide variety to divide into (that is, the biggest wide variety that could be a component of) each 2940 and 3150. In different words, it is the wide variety that carries all of the elements not unusual places to each number. In this case, the GCF is made from all of the elements that 2940 and 3150 have in a not unusual place.
Looking at the pleasant neat list, I can see that the numbers each have a component of 2; 2940 has a 2d replica of component 2, however, 3150 does now no longer, so I can most effective matter the only replica closer to my GCF. The numbers additionally proportion one replica of 3, one replica of 5, and one replica of 7.
2940: 2×2×3 ×5 ×7×7 3150: 2 ×3×3×5×5×7 —-:—————- GCF: 2 ×3 ×5 ×7 = 210
Then the GCF is 2 × 3 × 5 × 7 = 210.
On the alternative hand, the Least Common Multiple, the LCM, is the smallest (“least”) wide variety that each 2940 and 3150 will divide into. That is, it’s far the smallest wide variety that carries each 2940 and 3150 as elements, the smallest wide variety that could be a multiple of each of those values; it’s far the multiple not unusual place to the 2 values. Therefore, it is going to be the smallest wide variety that carries each component in those numbers.
Looking lower back on the list, I see that 3150 has one replica of the component of 2; 2940 has copies. Since the LCM need to include all elements of every wide variety, the LCM needs to include each copy of 2. However, to keep away from overduplication, the LCM does now no longer want 3 copies, due to the fact neither 2940 nor 3150 carries 3 copies.
This over-duplication problem with elements regularly reasons confusion, so permit’s spend a bit more time on this. Consider smaller numbers, 4 and 8, and their LCM. The wide variety 4 elements as 2 × 2; 8 elements as 2 × 2 × 2. The LCM desires most effectively have 3 copies of 2, to be able to be divisible through each 4 and 8. That is, the LCM is 8. You do now no longer want to take the 3 copies of 2 from the 8, after which throw in more copies from the 4. This could provide you 32. While 32 is a not unusual place multiple, due to the fact 4 and 8 each divide lightly into 32, 32 isn’t always the LEAST (smallest) not unusual place multiple, due to the fact you would have over-duplicated the 2s while you threw withinside the more copies from the 4.
Let me pressure again: permit the pleasant neat list to preserve the tune of factors for you, particularly whilst the numbers get big. Returning to the exercise: