Example 1
Use Euclid's algorithm to find the HCF of 135 and 225.
- 225 = 135 × 1 + 90.
- 135 = 90 × 1 + 45.
- 90 = 45 × 2 + 0.
- The last non-zero remainder is 45.
Answer: 45
♾️ Class 10 · Number system
Real Numbers for Class 10: notes, worked examples and practice
Real numbers bring together rational and irrational numbers. In Class 10 you learn the algorithm behind HCF, the theorem that every number has a unique prime factorisation, and how to tell terminating decimals from non-terminating ones.
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Notes
Real Numbers: what you need to know
Chapter names and order differ by board and textbook edition, so follow your own textbook for the order.
Euclid's division lemma
For any positive integers a and b there are unique whole numbers q and r with a = bq + r and 0 ≤ r < b.
Euclid's algorithm for HCF
Divide the larger by the smaller, then divide the divisor by the remainder, and repeat. The last non-zero remainder is the HCF.
Fundamental theorem of arithmetic
Every composite number can be written as a product of primes in exactly one way (apart from the order). It gives HCF and LCM from prime factors.
Rational and irrational numbers
A rational number can be written as p/q. An irrational number, such as √2, √3 or π, has a decimal expansion that never ends and never repeats.
Terminating decimals
A rational number p/q (in lowest terms) has a terminating decimal if and only if the prime factors of q are only 2 and 5. 7/40 terminates because 40 = 2³ × 5.
Formulas and rules to remember
- a = bq + r, 0 ≤ r < b
- HCF × LCM = product of the two numbers
- p/q terminates iff q = 2ᵐ × 5ⁿ
Solved examples
Example 2
Find the HCF and LCM of 96 and 404 using prime factorisation.
- 96 = 2⁵ × 3 and 404 = 2² × 101.
- HCF = 2² = 4.
- LCM = 2⁵ × 3 × 101 = 9696.
Answer: HCF 4, LCM 9696
Example 3
Will 7/40 have a terminating decimal? Write it.
- 40 = 2³ × 5, so the only prime factors are 2 and 5.
- So it terminates: 7 ÷ 40 = 0.175.
Answer: 0.175
Example 4
Will 13/125 have a terminating decimal? Write it.
- 125 = 5³, so the only prime factor is 5.
- So it terminates: 13 ÷ 125 = 0.104.
Answer: 0.104
Avoid these
Common mistakes
Stopping Euclid's algorithm early
Continue until the remainder is zero. The HCF is the last non-zero remainder.
Thinking a long decimal means irrational
1/7 = 0.142857… repeats, so it is rational. Irrational decimals never repeat.
Forgetting to put the fraction in lowest terms
Reduce p/q first, then look at the prime factors of q.
Assuming the sum of irrationals is irrational
√2 + (−√2) = 0, which is rational.
10 practice questions with answers
Try each question on paper first, then tap it to check your answer. The questions are generated by code and each answer is checked.
The LCM of 21 and 19 is
Answer: 399
The HCF (GCF) of 54 and 23 is
Answer: 1
The HCF (GCF) of 41 and 25 is
Answer: 1
The HCF (GCF) of 19 and 38 is
Answer: 19
The HCF (GCF) of 35 and 28 is
Answer: 7
The HCF (GCF) of 59 and 5 is
Answer: 1
√529 =
Answer: 23
√225 =
Answer: 15
√441 =
Answer: 21
√324 =
Answer: 18
More practice
Printable worksheets (PDF with answers)
Games and lessons
LCM & HCF calculator
Find the lowest common multiple and highest common factor (GCD) of up to eight numbers, with a prime factor table that shows why.
Use the tool →Prime number checker
Check whether a number is prime, see its prime factorisation, all its factors, the nearest primes, or list every prime up to 100,000.
Use the tool →FAQ
Real Numbers Class 10: questions students ask
What is Euclid's division algorithm?
A method for finding the HCF by repeated division with remainders. The last non-zero remainder is the HCF.
What is the fundamental theorem of arithmetic?
Every composite number is a product of primes in a unique way, apart from the order of the factors.
How do I know if a fraction has a terminating decimal?
Reduce it to lowest terms. If the denominator has no prime factors other than 2 and 5, the decimal terminates.
What is an irrational number?
A number that cannot be written as p/q, such as √2 or π. Its decimal never ends and never repeats.
Which board is this real numbers page for?
It covers the ideas taught in CBSE, ICSE and state-board maths. Chapter names and order differ by board and textbook edition, so follow your own textbook for the exact order.
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