Fundamental Theorem of Arithmetic (Applications) MCQs Quiz | Class 10

Fundamental Theorem of Arithmetic: Deep Dive

The Fundamental Theorem of Arithmetic is a cornerstone of number theory, providing a unique way to understand composite numbers. It states that every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur. Let us explore its concepts and applications, particularly in finding HCF and LCM.

Key Concepts

• Prime Numbers: A natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11…).
• Composite Numbers: A natural number greater than 1 that is not prime; it has at least one divisor other than 1 and itself (e.g., 4, 6, 8, 9, 10…).
• Fundamental Theorem of Arithmetic: Also known as the unique factorization theorem, it asserts that any integer greater than 1 can be written as a product of prime numbers, and this product is unique up to the order of the factors. For example, 36 = 2 x 2 x 3 x 3 = 2^2 x 3^2. There is no other combination of prime numbers that will multiply to give 36.

Applications: HCF and LCM via Prime Factorization

The Fundamental Theorem of Arithmetic is extremely useful for finding the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two or more integers.

Finding HCF (Highest Common Factor)

To find the HCF of two or more numbers using prime factorization, we take the product of the smallest power of each common prime factor involved in the numbers.

Example: Find the HCF of 12 and 18.

• Prime factorization of 12 = 2 x 2 x 3 = 2^2 x 3^1
• Prime factorization of 18 = 2 x 3 x 3 = 2^1 x 3^2
• Common prime factors are 2 and 3.
• Smallest power of 2 is 2^1.
• Smallest power of 3 is 3^1.
• HCF(12, 18) = 2^1 x 3^1 = 6.

Finding LCM (Least Common Multiple)

To find the LCM of two or more numbers using prime factorization, we take the product of the highest power of all prime factors involved in the numbers.

Example: Find the LCM of 12 and 18.

• Prime factorization of 12 = 2^2 x 3^1
• Prime factorization of 18 = 2^1 x 3^2
• All prime factors involved are 2 and 3.
• Highest power of 2 is 2^2.
• Highest power of 3 is 3^2.
• LCM(12, 18) = 2^2 x 3^2 = 4 x 9 = 36.

Relation between HCF and LCM

For any two positive integers ‘a’ and ‘b’, the product of their HCF and LCM is equal to the product of the numbers themselves.
HCF(a, b) x LCM(a, b) = a x b
For our example: HCF(12, 18) x LCM(12, 18) = 6 x 36 = 216. And 12 x 18 = 216. The relation holds true.

Concept-based Questions (Revisited)

• Numbers ending with digit zero: A number will end with the digit zero if and only if its prime factorization includes both 2 and 5. For example, 10 = 2 x 5, 20 = 2^2 x 5. If a number lacks either 2 or 5 in its prime factors, it cannot end with a zero.
• Identifying composite numbers: An expression like 7 x 11 x 13 + 13 can be identified as composite because you can factor out 13, showing it has factors other than 1 and itself (e.g., 13 x (7 x 11 + 1) = 13 x 78).
• Co-prime numbers: Two numbers are said to be co-prime (or relatively prime) if their HCF is 1. This means they share no common prime factors. For example, 4 and 9 are co-prime (HCF(4,9)=1).

Quick Revision Points

• Every composite number has a unique prime factorization.
• HCF is the product of the smallest powers of common prime factors.
• LCM is the product of the highest powers of all prime factors.
• For any two numbers ‘a’ and ‘b’, HCF(a, b) x LCM(a, b) = a x b.
• A number ends with zero if its prime factorization includes both 2 and 5.

Further Practice Questions

1. Check whether 4^n can end with the digit zero for any natural number n.
2. Explain why 3 x 5 x 7 + 7 is a composite number.
3. Given that HCF(96, 404) = 4, find the LCM(96, 404).
4. Find the prime factorization of 156.
5. Can two numbers have 18 as their HCF and 380 as their LCM? Give reason.