Fundamental Theorem of Arithmetic: Class 10
Learn prime factorisation, HCF and LCM word problems, and why HCF × LCM equals the product of two positive integers.
The fundamental theorem of arithmetic says that an integer greater than 1 has one prime factorisation, apart from rearranging the factors. A prime is greater than 1 and has exactly two positive divisors. The theorem lets us compare numbers prime by prime instead of listing every divisor or multiple.
How to find HCF and LCM by prime factorisation
Prime factorise 156
- 1156 ÷ 2 = 78, 78 ÷ 2 = 39, 39 ÷ 3 = 13.
- 213 is prime, so 156 = 2² × 3 × 13. Multiply back to check.
Compare 72 and 120
- 172 = 2³ × 3²; 120 = 2³ × 3 × 5.
- 2HCF = 2³ × 3 = 24. Only common primes with the smaller exponents can divide both.
- 3LCM = 2³ × 3² × 5 = 360. A common multiple must contain enough of every prime for both numbers.
Why HCF × LCM equals the product
For a particular prime, suppose its exponents in a and b are m and n; use exponent zero if the prime is absent. Its exponent in HCF(a,b) is min(m,n), and in LCM(a,b) it is max(m,n). Adding the smaller and larger exponent gives m + n, exactly its exponent in ab. This holds for every prime, so the products are equal for two positive integers.
Numerical check
- 124 × 360 = 8640 and 72 × 120 = 8640.
- 2The identity does not extend unchanged to three numbers. For 2, 4 and 8, HCF × LCM = 16, while their product is 64.
HCF and LCM word problems
Cut 48 cm and 60 cm ribbons into equal longest pieces, with none left
- 1The piece length must divide both ribbon lengths, so use HCF(48,60) = 12 cm.
- 2The ribbons give 4 and 5 pieces respectively. The length is 12 cm; the total number of pieces is 9.
Two events repeat every 8 and 12 minutes and happen together now
- 1The next shared time must be a multiple of both intervals.
- 2LCM(8,12) = 24 minutes. The first event repeats 3 times and the second 2 times by then.
Check your understanding
- 1. HCF and LCM of 45 and 60? Answer: 15 and 180.
- 2. Why is 1 not included as a prime factor? Answer: it has only one positive divisor and does not change a product.
- 3. Does “greatest” always mean HCF? No. Read what the quantity must do: a divisor fits into each amount; a multiple contains each amount a whole number of times.