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Fundamental theorem of arithmetic claims that, “factorization of every composite number can be written as a product of primes regardless of the sequence in which the prime factors of that individual number occur.” The arithmetic fundamental theorem is a very useful way of understanding the prime factorization of any number.
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Key Terms: Fundamental Theorem of Arithmetic, HCF, LCM, Prime Factorization, Composite Number, Prime Number
Fundamental Theorem of Arithmetic
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Fundamental theorem of Arithmetic claims, "factorization of every composite number can be written as a product of primes regardless of the sequence in which the prime factors of that number occur."
It claims that each integer greater than one is either a prime number or can be written as a prime number. In other words, all natural numbers can be represented as the product of their prime components. To recap, prime factors are numbers that are only divisible by 1 and themselves.
The number 35, for example, can be expressed in the form of its prime factors as: Prime factors of 35 are 7 and 5.
Similarly, using the prime factorization method, another number 114560 can be expressed as the product of its prime factors.
| 114560 = 27 × 5 × 179 |
As a result, we factored 114560 as the product of its primes’ power. As a result, every natural number may be written as the product of the powers of its primes. The Fundamental Theorem of Arithmetic, unique factorization theorem, or unique-prime-factorization theorem is the name given to this proposition.
Fundamental Theorem of Arithmetic
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Fundamental Theorem of Arithmetic Detailed Video Explanation:
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Proof for Fundamental Theorem of Arithmetic
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To prove the fundamental theorem of arithmetic, we must show that the prime factorisation exists and is unique. As a result, the fundamental theorem of arithmetic states that proof consists of two steps.
We will prove that the product of primes can only be represented in one way for every integer, n2.
| N = p1.p2 … pi |
First, determine whether prime factorisation exists in the first place.This will be demonstrated via mathematical induction. Mathematical induction is a method for proving that a statement, formula, or theorem holds true for all natural numbers. As indicated below, the technique entails two steps to prove a statement.
Base step
It establishes that a statement is true for the initial value in the first step. The assertion is correct for n = 2.
Inductive step
It establishes that if a statement is true for the nth iteration (or number n), it is also true for the (n+1)th iteration (or number (n+1)th iteration or number (n+1). Assume that the assertion for n = k is correct.
As a result, the product of primes can be written as k. Let's see if the proposition is true for n = k+1.
- If k+1 is prime, the case is obvious.
- If k+1 isn't prime, there's a good chance it has a prime factor, like p. Then k + 1 = pj, where jk is a constant (1)
The inductive step can be used to express jk, k as the product of primes. K+1 can also be stated as a prime product as a result of (1). Mathematical induction is used to prove the presence of factorisation.
Uniqueness of Prime FactorisationAssume that n may be written as a product of primes in two ways, for example, ⇒ n = p1p2…pi = q1q2 … qj Because these are prime factorisations, “q1q2...qj” are coprime numbers (as they are prime numbers). Euclid's Lemma states that p1 divides just one of the primes as a result. As q1 is the smallest prime, p1=q1. Similarly, for all n, we may prove that pn=qn. As a result, i=j is obtained. As a result, n′ prime factorization is unique. |
The video below explains this:
Arithmetic Progression Detailed Video Explanation:
HCF and LCM Using Fundamental Theorem of Arithmetic
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The fundamental theorem of arithmetic is used to calculate the HCF and LCM of two numbers. To do so, we must first determine the prime factors of both numbers. After that, we’ll have a look at the following:
- HCF is the product of each common prime factor’s smallest power.
- LCM is the product of each common prime factor’s highest power.
- Let’s look at the HCF of 850 and 680, for example. We’ll start by determining the prime factorization of these numbers.
- Prime factorization of 850 = 21 × 52 × 171
- Prime factorization of 680 = 23 × 51 × 171
- HCF is the product of the smallest power of every common prime factor. As a result, HCF (850, 680) = 21 × 51 × 171 = 170.
- LCM is the product of the greatest power of every common prime factor. As a result, LCM (850, 680) = 23 × 52 × 171 = 3400.
Thus,
→ HCF (850, 680) = 170
→ LCM (850, 680) = 3400
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Things To Remember
- According to the Fundamental Theorem of Arithmetic, any composite number can be factored in as a product of primes
- Carl Friedrich Gauss discovered the fundamental theorem of Arithmetic in 1801.
- The fundamental theorem of arithmetic guarantees the existence and uniqueness of the prime factorisation of a number
- The fundamental theorem of arithmetic is used to calculate the HCF and LCM.
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Sample Questions
Ques 1: Using the fundamental theorem of arithmetic, write 1080 as the product of prime factors. (2 Marks)
Ans. We know that we can write 1080 as a product of its prime factors using the fundamental theorem of arithmetic. The prime factorization of 1080 will be determined as:
Prime factors of 1080 = 2 × 2 × 2 × 3 × 3 × 3 × 5
= 23 × 33 × 51
Therefore, 23 × 33 × 51 is the prime factorization of 1080.
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Ques 2: In a formula racing competition, two racing vehicles A and B take 30 minutes and 45 minutes to complete one round of the track, respectively. How long will it take for the cars to reunite at the starting point? (2 Marks)
Ans. As car B takes longer to complete one circle than car A, it’s safe to infer that A will arrive first, and the two cars will meet again after A has arrived at the starting place. Finding the L.C.M of the time taken by each can be used to compute this time.
30 = 2 × 3 × 5
45 = 3 × 3 × 5
The L.C.M is 90.
Thus, both cars will reunite at the starting point after 90 minutes.
Ques 3: Find the HCF of 126,162 and 180 by using the fundamental theorem of arithmetic. (4 Marks)
Ans. Let us find the prime factorisations of 126,162,and180
Thus, 126,=2×3×3×7=21×32×71
162=2×3×3×3×3=21×34
180=2×2×3×3×5=22×32×51
The HCF of two or more numbers is the smallest power of each common prime factor in the numbers.
Hence, HCF(126,162,180)=21×32=18
Ques 4: Find the HCF and LCM of 26 and 91 and Prove that LCM × HCF = Product of Two Numbers. (3 Marks)
Ans. By prime factorization
26 = 2 x 13
91 = 7 x 13
HCF (26, 91) = 13
LCM (26, 91) = 13 x 2 x 7
= 26 × 7 = 182
LCM × HCF = 13 × 182 = 2366
Product of two numbers = 26 × 91 = 2366.
Hence, L.C.M. × H.C.F. = Product of two numbers.
Ques 5: Find the LCM and HCF of numbers 6 and 20. (4 Marks)
Ans. Prime Factorization of 6 can be represented in the following way,
Prime Factorization of 20 can be represented in the following way,
So, now we have prime factorization of both the numbers,
6 = 2 × 3
20 = 2 × 2 × 5
We know that
HCF = Product of the smallest power of each common prime factor in the numbers.
LCM = Product of the greatest power of each prime factor, involved in the numbers.
So, HCF(6,20) = 21
LCM(6,20) = 22 × 31 × 5
Ques 6: Suppose that for two numbers “a” and “b”. HCF is given which is 120 and the product of the two numbers is given as 3600. Find the LCM of the two numbers. (2 Marks)
Ans. Given two numbers “a” and “b”.
LCM(a, b) is unknown while HCF(a, b) = 120 and a × b = 3600.
From the property studied above,
HCF(a, b) × LCM(a, b) = a × b
Plugging in the given values.
120 × LCM(a, b) = 3600
LCM(a, b) = 30
Ques 7: For what values of natural number n, 4n can end with the digit 6? (2 Marks)
Ans.
if n = 1, then 41 = 4
if n = 2, then 42 = 16
if n = 3, then 43 = 64
if n = 4, then 44 = 256
if n = 5, then 45 = 1024
if n = 6, then 46 = 4096
When n is even, 4n ends with 6.
Ques 8: If m, n are natural numbers, for what values of m, does 2n x 5m ends in 5? (2 Marks)
Ans. For any value of n, 2n will become even.
For any value m, 5m ends with 5. The product of even number and a number ends with the digit 5, we get the answer ends with 0.
We should not apply the value 0 for n and m, because n and m are natural numbers.
Ques 9: Find the least number that is divisible by the first ten natural numbers. (2 Marks)
Ans. First 10 natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
L.C.M of these natural numbers:
Factors = 2 × 2 × 2 × 3 × 3 × 5 × 7
= 2520
The smallest number is 2520.
Ques 10: Find the greatest number consisting of 6 digits which is exactly divisible by 24,15,36? (3 Marks)
Ans. We will find out the LCM of 24,15 and 36 is 360.
The greatest 6-digit number is 999999.
Now, We will divide this number by LCM of 24,15 and 36, we will get,
999999/360 will get the remainder 279.
Now 999999 – 279 = 999720.
Now we will check it out for the numbers we will get,
999720/24 = 41655
999720/15 = 66648
999720/36 = 27770
999720 is the greatest number 6-digit number divisible by 24,15 and 36.
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