Breaking Down HCF and LCM of 90 and 36 Using First Principles

Let's break down the numbers 90 and 36 into their prime factors.Starting with 90, we first divide by 2, which gives us 45.Next, 45 divides by 3 to give us 15.Finally, 15 breaks down into 3 times 5, both prime numbers.Now for 36, we start by dividing by 2, getting 18.18 divides by 2 again to give us 9.Finally, 9 breaks down into 3 times 3.Now we can write these as prime factorizations. For 90, we have two cubed times three squared times five. For 36, we have two cubed times three squared.Now that we have our prime factorizations, let's find the Highest Common Factor.Let's organize our prime factors in a table to compare them.For each prime factor, we'll take the lowest power in which it appears in either number.For our HCF, we only include prime factors that appear in both numbers.We take 2 to the first power, since that's the lower power between 90 and 36.And 3 squared appears in both numbers, so we include that as well.Now let's calculate: 2 times 3 squared, which is 2 times 9, equals 18.Let's verify that 18 is indeed the Highest Common Factor.90 divided by 18 equals 5, with no remainder.And 36 divided by 18 equals 2, also with no remainder.Therefore, 18 is confirmed as the Highest Common Factor of 90 and 36.To find the Least Common Multiple, we'll use the prime factorizations we found earlier.The LCM uses the highest power of each prime factor that appears in either number.Let's compare the powers of each prime factor in both numbers.For our LCM, we'll use two cubed, which is the highest power of two, three squared, which appears in both numbers, and five to the first power.Let's expand this to see the actual multiplication.Two cubed is eight, three squared is nine, and we multiply by five.This gives us three hundred and sixty as our Least Common Multiple.Let's verify that three hundred and sixty is indeed our LCM by dividing it by both ninety and thirty-six.Three hundred and sixty divided by ninety equals four, showing that ninety goes into it evenly.And three hundred and sixty divided by thirty-six equals ten, confirming it's also divisible by thirty-six.