Real Numbers | Exercise 1.1

Question 1

Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Solution

Understand the Question

To express a number as a product of its prime factors, divide the number successively by the smallest possible prime numbers ( $2, 3, 5, 7, 11, 13, 17, \dots$ ) until the quotient becomes $1$ .
Write repeated prime factors in exponential form (e.g., $2 \times 2 = 2^2$ ).

(i) 140

Step 1 · Prime Factorise 140

Diagram 1

\begin{aligned} 140 \div 2 &= 70 \\ 70 \div 2 &= 35 \\ 35 \div 5 &= 7 \\ 7 \div 7 &= 1 \end{aligned}

140 = 2 \times 2 \times 5 \times 7 = 2^2 \times 5 \times 7

Answer

(i) $2^2 \times 5 \times 7$

(ii) 156

Step 1 · Prime Factorise 156

Diagram 2

\begin{aligned} 156 \div 2 &= 78 \\ 78 \div 2 &= 39 \\ 39 \div 3 &= 13 \\ 13 \div 13 &= 1 \end{aligned}

156 = 2 \times 2 \times 3 \times 13 = 2^2 \times 3 \times 13

Answer

(ii) $2^2 \times 3 \times 13$

(iii) 3825

Step 1 · Prime Factorise 3825

\begin{aligned} 3825 \div 3 &= 1275 \\ 1275 \div 3 &= 425 \\ 425 \div 5 &= 85 \\ 85 \div 5 &= 17 \\ 17 \div 17 &= 1 \end{aligned}

3825 = 3 \times 3 \times 5 \times 5 \times 17 = 3^2 \times 5^2 \times 17

Answer

(iii) $3^2 \times 5^2 \times 17$

(iv) 5005

Step 1 · Prime Factorise 5005

\begin{aligned} 5005 \div 5 &= 1001 \\ 1001 \div 7 &= 143 \\ 143 \div 11 &= 13 \\ 13 \div 13 &= 1 \end{aligned}

5005 = 5 \times 7 \times 11 \times 13

Answer

(iv) $5 \times 7 \times 11 \times 13$

(v) 7429

Step 1 · Prime Factorise 7429

\begin{aligned} 7429 \div 17 &= 437 \\ 437 \div 19 &= 23 \\ 23 \div 23 &= 1 \end{aligned}

7429 = 17 \times 19 \times 23

Answer

(v) $17 \times 19 \times 23$

Common Mistakes

Dividing by Composite Numbers: Dividing by composite numbers like $4, 6, 9$ instead of prime numbers.
Testing Larger Primes: For numbers like $7429$ , which have no small prime factors ( $2, 3, 5, 7, 11, 13$ ), test systematically with larger prime numbers like $17, 19, 23$ .
Exponential Notation: Forgetting to write repeated factors in exponential form (e.g., writing $2 \times 2$ instead of $2^2$ ).

More questions in Exercise 1.1