LCM and HCF of 36 and 90 Explained with Animations

Let's learn how to break down numbers into their prime factors.We'll work with the numbers 36 and 90.Let's start with 36. We'll divide by the smallest prime number that goes in evenly.Now let's do the same with 90.Let's collect all the prime factors for 36.And now for 90.For 36, we can write this as 2 squared times 3 squared.And for 90, we have 2 times 3 squared times 5.Now that we have our prime factorizations, let's find the Highest Common Factor.Let's compare the prime factors and their powers in both numbers.For the number 2, it appears as 2 squared in 36, but only as 2 to the first power in 90. We take the minimum power, which is 1.For 3, it appears as 3 squared in both numbers, so we take the power of 2.The number 5 only appears in 90, so it won't be part of our Highest Common Factor.Therefore, our Highest Common Factor will use 2 to the first power and 3 squared.This gives us 2 times 9Which equals 18To verify this, let's list out all the factors of both numbers.Here are all the factors of 90.Looking at the common factors between these numbers, we can confirm that 18 is indeed the highest common factor.Now that we've found the Highest Common Factor, we can move on to finding the Least Common Multiple.To find the Least Common Multiple, we'll use the prime factorizations we found earlier.The LCM is found by taking each prime factor with its highest power from either number.Let's compare the powers of each prime factor in both numbers.For 2, the highest power is 2 from thirty-six.For 3, both numbers have power 2.And 5 appears only in ninety with power 1.Therefore, our LCM will use 2 squared, 3 squared, and 5 to the first power.Let's calculate: 2 squared is 4, 3 squared is 9, and 5 remains 5.Multiplying these numbers gives us 180.We can verify that 180 is indeed the least common multiple by dividing it by both numbers.180 divided by 36 equals 5And 180 divided by 90 equals 2, confirming that both numbers divide evenly into 180.