Doubtclip is an AI tutor that turns any homework or exam doubt into a step-by-step whiteboard video lesson with diagrams, formulas, and voice explanation.

Who is the founder of Doubtclip?

Vinay Malav is the founder of Doubtclip (Founder, Doubtclip, based in Kota, Rajasthan, India). More: https://www.doubtclip.com/about

Which subjects and exams does it support?

It supports Physics, Chemistry, Mathematics/Calculus and related STEM doubts for school boards, JEE, NEET, and global high-school / undergrad practice.

How do I get a video solution?

Upload a photo of the question or paste the text, sign in, and Doubtclip generates a whiteboard-style explanation video you can watch and share.

Is Doubtclip available worldwide?

Yes. Doubtclip is a web app for students globally. Lessons are available in English and Hinglish.

After how many decimal places will the decimal expansion of 23 / (2⁴ × 5³) terminate?

4 places. The decimal terminates after max(n, m) places. Here powers are 4 and 3; max(4, 3) = 4 places.

Mathematics · Class 9-10 · CBSE

Real Numbers: Prime Factorisation and Irrationality Proofs

Master the Fundamental Theorem of Arithmetic, prime factorisation method for HCF and LCM, proving the irrationality of √2, √3, and √5 by contradiction, and decimal expansion conditions for rational numbers.

1Fundamental Theorem of Arithmetic and HCF-LCM Relations

TheFundamental Theorem of Arithmeticstates that every composite number can be expressed (factorised) uniquely as a product of prime numbers, apart from the order in which the prime factors occur.

Product Property for Two Positive Integers

\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b

Prime Factorisation Definitions: HCF vs LCM

Concept	Prime Factor Definition	Calculation Procedure	Example for 36 and 84
HCF (Highest Common Factor)	Product of the SMALLEST power of each common prime factor involved in the numbers	36 = 2² × 3²; 84 = 2² × 3 × 7. Common primes: 2 (min power 2), 3 (min power 1)	HCF = 2² × 3 = 12
LCM (Lowest Common Multiple)	Product of the GREATEST power of each prime factor involved in the numbers	Highest powers: 2², 3², 7¹	LCM = 2² × 3² × 7 = 4 × 9 × 7 = 252

HCF (Highest Common Factor)
Prime Factor Definition
Product of the SMALLEST power of each common prime factor involved in the numbers
Calculation Procedure
36 = 2² × 3²; 84 = 2² × 3 ×
7. Common primes: 2 (min power 2), 3 (min power 1)
Example for 36 and 84
HCF = 2² × 3 = 12
LCM (Lowest Common Multiple)
Prime Factor Definition
Product of the GREATEST power of each prime factor involved in the numbers
Calculation Procedure
Highest powers: 2², 3², 7¹
Example for 36 and 84
LCM = 2² × 3² × 7 = 4 × 9 × 7 = 252

2Proof of Irrationality by Contradiction

A number isIrrationalif it cannot be written in the form p/q where p, q are integers and q ≠
0. We prove numbers like √2, √3, √5 are irrational using the technique ofProof by Contradiction.

Step-by-Step Proof that √2 is Irrational

1
1. Assumption: Assume to the contrary that √2 is rational. Hence √2 = a/b, where a and b are co-prime integers (HCF = 1) and b ≠ 0.
2
2. Squaring both sides: 2 = a²/b² ⇒ 2b² = a². Therefore, 2 divides a². By Theorem, if a prime p divides a², then p divides a. Hence 2 divides a.
3
3. Substitute a = 2c: 2b² = (2c)² = 4c² ⇒ b² = 2c². This implies 2 divides b²; hence 2 divides b.
4
4. Contradiction: 2 divides both a and b. This contradicts our initial premise that a and b are co-prime (have no common factor other than 1).
5
5. Conclusion: Our assumption that √2 is rational was false. Therefore, √2 is strictly irrational.

3Decimal Expansions of Rational Numbers

Terminating vs Non-Terminating Recurring Decimals

Decimal Form	Condition on Denominator q (in simplest p/q form)	Decimal Behavior	Example
Terminating Decimal	Prime factorisation of q is strictly of the form 2ⁿ · 5ᵐ (where n, m are non-negative integers)	Ends after finite digits	13/80 = 13/(2⁴ × 5¹) = 0.1625
Non-Terminating Repeating	Prime factorisation of q contains prime factors other than 2 and 5 (e.g. 3, 7, 11)	Infinite repeating block of digits	17/30 = 17/(2 × 3 × 5) = 0.5666...

Terminating Decimal
Condition on Denominator q (in simplest p/q form)
Prime factorisation of q is strictly of the form 2ⁿ · 5ᵐ (where n, m are non-negative integers)
Decimal Behavior
Ends after finite digits
Example
13/80 = 13/(2⁴ × 5¹) = 0.1625
Non-Terminating Repeating
Condition on Denominator q (in simplest p/q form)
Prime factorisation of q contains prime factors other than 2 and 5 (e.g. 3, 7, 11)
Decimal Behavior
Infinite repeating block of digits
Example
17/30 = 17/(2 × 3 × 5) = 0.5666...

4Exam Corner: Tips, Mistakes & Board Questions

Critical Board Exam Takeaways

The relation HCF(a, b) × LCM(a, b) = a × b holds true ONLY for two numbers; it is FALSE for three or more numbers.
The sum, difference, product, or quotient of a non-zero rational number and an irrational number is always irrational (e.g. 3 + 2√5 is irrational).
To find whether p/q terminates, always reduce the fraction to simplest lowest terms first so that p and q are co-prime.

Exam question · CBSE Class 10 Board

If HCF(306, 657) = 9, what is LCM(306, 657)?

A.22338
B.21456
C.23450
D.20198

Remember

Every composite number has a uniqueprime factorisation.
For any two positive integers: ==HCF(a, b) × LCM(a, b) = a × b==.
Irrationality of $\sqrt{2}, \sqrt{3}, \sqrt{5}$ is established viacontradiction.
A rational fraction $p/q$ terminates if and only if == $q = 2^n \cdot 5^m$ ==.

Check yourself

Question 1
After how many decimal places will the decimal expansion of 23 / (2⁴ × 5³) terminate?