HCF of 144 and 180 is expressed in the form 13m – 16. Find the value of m.

To find the value of m in the expression for the highest common factor (HCF) of 144 and 180, we first need to calculate the HCF itself.

Calculating HCF

The HCF of two numbers can be found using their prime factorization or the Euclidean algorithm. Here, we'll use the prime factorization method:

• 144: 2 × 2 × 2 × 2 × 3 × 3 = 2⁴ × 3²
• 180: 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5

Finding Common Factors

The common prime factors are:

• For 2: the minimum power is 2 (from 180).
• For 3: the minimum power is 2 (from both).

Calculating HCF

Thus, the HCF is:

HCF = 2² × 3² = 4 × 9 = 36

Setting Up the Equation

Now, we know that the HCF can be expressed as:

36 = 13m - 16

Solving for m

To find m, we rearrange the equation:

13m = 36 + 16

13m = 52

Now, divide both sides by 13:

m = 52 / 13 = 4

Final Answer

The value of m is 4.