Understanding HCF (Highest Common Factor)

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. To find the HCF of 24 and 144, we can use methods like prime factorization or the division method. Let's use the prime factorization method.

Finding the HCF of 24 and 144 using Prime Factorization

First, we find the prime factorization of each number:

• Prime factorization of 24: $$24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1$$
• Prime factorization of 144: $$144 = 12 \times 12 = (2^2 \times 3) \times (2^2 \times 3) = 2^{2+2} \times 3^{1+1} = 2^4 \times 3^2$$

To find the HCF, we take the common prime factors with the lowest power they appear in either factorization.

• Common prime factors are 2 and 3.
• Lowest power of 2 is $2^3$.
• Lowest power of 3 is $3^1$.

So, the HCF of 24 and 144 is: $$\text{HCF}(24, 144) = 2^3 \times 3^1 = 8 \times 3 = 24$$

Finding the Value of 'p'

We are given that the HCF of 24 and 144 is equal to '10p + 4'. We have found that the HCF is 24.

Therefore, we can set up the equation: $$10p + 4 = 24$$

Now, we need to solve this linear equation for 'p'.

1. Subtract 4 from both sides of the equation: $$10p + 4 - 4 = 24 - 4$$ $$10p = 20$$
2. Divide both sides by 10: $$\frac{10p}{10} = \frac{20}{10}$$ $$p = 2$$

Thus, the value of 'p' is 2.

Revision Table: HCF and Related Concepts

Additional Information: Solving Linear Equations

A linear equation in one variable is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants and $a \neq 0$. To solve for the variable (like 'p' in our problem), we use inverse operations to isolate the variable on one side of the equation.

• If a number is added to the term with the variable, subtract that number from both sides.
• If a number is subtracted from the term with the variable, add that number to both sides.
• If the variable is multiplied by a number, divide both sides by that number.
• If the variable is divided by a number, multiply both sides by that number.

In the equation $10p + 4 = 24$, we first subtracted 4 (inverse of adding 4) and then divided by 10 (inverse of multiplying by 10) to find the value of 'p'.