Grade 7 HCF and LCM

Step 1: Find the prime factorization of each number.

• Prime factors of 24: 24 = 2 x 12 = 2 x 2 x 6 = 2 x 2 x 2 x 3 = 2 to the power of 3 x 3 to the power of 1
• Prime factors of 36: 36 = 2 x 18 = 2 x 2 x 9 = 2 x 2 x 3 x 3 = 2 to the power of 2 x 3 to the power of 2

Step 2: Identify the common prime factors.

The common prime factors are 2 and 3.

Step 3: Multiply the common prime factors, taking the lowest power of each common factor.

• Lowest power of 2: 2 to the power of 2 (from 2 to the power of 3 and 2 to the power of 2)
• Lowest power of 3: 3 to the power of 1 (from 3 to the power of 1 and 3 to the power of 2)

HCF = 2 to the power of 2 x 3 to the power of 1 = 4 x 3 = 12.

Step 4: The HCF of 24 and 36 is 12.

Step 1: Divide the larger number (108) by the smaller number (84).

108 = 84 x 1 + 24 (Remainder is 24)

Step 2: Take the remainder (24) as the new divisor and the previous divisor (84) as the new dividend.

84 = 24 x 3 + 12 (Remainder is 12)

Step 3: Take the new remainder (12) as the new divisor and the previous divisor (24) as the new dividend.

24 = 12 x 2 + 0 (Remainder is 0)

Step 4: The last non-zero divisor is the HCF.

The last divisor was 12.

Step 5: The HCF of 84 and 108 is 12.

Step 1: Find the prime factorization of each number.

• Prime factors of 15: 15 = 3 x 5 = 3 to the power of 1 x 5 to the power of 1
• Prime factors of 25: 25 = 5 x 5 = 5 to the power of 2

Step 2: Identify all prime factors (common and uncommon).

The prime factors are 3 and 5.

Step 3: Multiply all the prime factors, taking the highest power of each factor.

• Highest power of 3: 3 to the power of 1
• Highest power of 5: 5 to the power of 2

LCM = 3 to the power of 1 x 5 to the power of 2 = 3 x 25 = 75.

Step 4: The LCM of 15 and 25 is 75.

Step 1: Write the numbers in a row and divide by the smallest prime that divides at least one number.

2 | 12, 18, 27
  |  6,  9, 27  (2 divides 12 and 18, 27 is brought down)
                                    

Step 2: Continue dividing until all numbers are 1.

2 | 12, 18, 27
3 |  6,  9, 27
3 |  2,  3,  9
2 |  2,  1,  3
3 |  1,  1,  1
  |  1,  1,  1
                                    

Step 3: Multiply all the divisors to get the LCM.

Divisors are 2, 3, 3, 2, 3.

LCM = 2 x 3 x 3 x 2 x 3 = 108.

Step 4: The LCM of 12, 18, and 27 is 108.

Step 1: Find the HCF of 12 and 18.

From previous examples or calculation, HCF(12, 18) = 6.

Step 2: Find the LCM of 12 and 18.

From previous examples or calculation, LCM(12, 18) = 36.

Step 3: Calculate the product of HCF and LCM.

HCF x LCM = 6 x 36 = 216.

Step 4: Calculate the product of the numbers themselves.

Product of numbers = 12 x 18 = 216.

Step 5: Compare the results.

Since 216 = 216, the relationship HCF(12, 18) multiplied by LCM(12, 18) = 12 multiplied by 18 is verified.