CBSE Class 10 Mathematics

Primes and composites

A prime number is a natural number greater than 1 whose only factors are 1 and itself. A composite number is a natural number greater than 1 that has at least one other factor, so it can be written as a product of smaller natural numbers. The number 1 is neither: it is left out of the primes so that factorisation is unique, because if 1 counted as a prime then $6 = 2 \times 3 = 1 \times 2 \times 3 = 1 \times 1 \times 2 \times 3$ would have infinitely many different factorisations. The number 2 is the only even prime, since every other even number has 2 as a factor.

Two positive integers are coprime (or relatively prime) when their only common factor is 1. The numbers 8 and 15 are coprime even though neither is prime, and two different primes are always coprime.

The fundamental theorem of arithmetic

Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

"Unique apart from the order" means that $360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$ can also be written $5 \times 3 \times 2 \times 3 \times 2 \times 2$, but no other collection of primes multiplies to 360. The usual way to write a factorisation is with the primes in increasing order and repeated primes as powers:

To find it, divide repeatedly by the smallest prime that goes in, starting with 2, and continue until the quotient is 1. For 2,310 the working is

so $2310 = 2 \times 3 \times 5 \times 7 \times 11$.

The theorem says two things, and each is used separately. Existence means a number can always be broken down into primes. Uniqueness means that if a prime divides a product, it must already appear in the factorisation of one of the factors, and that a number's prime factorisation can be used as an identifying fingerprint when comparing two numbers.

HCF and LCM by prime factorisation

The highest common factor (HCF) of two or more positive integers is the largest integer that divides each of them. The lowest common multiple (LCM) is the smallest positive integer that each of them divides.

Write each number as a product of prime powers. Then

• the HCF is the product of each prime that appears in every number, taken to the smallest power it has in any of them;
• the LCM is the product of every prime that appears in any of the numbers, taken to the greatest power it has in any of them.

For 180 and 168:

The primes 2 and 3 are common, with smallest powers $2^2$ and $3^1$, so $\text{HCF} = 2^2 \times 3 = 12$. Every prime that appears, with its greatest power, gives $\text{LCM} = 2^3 \times 3^2 \times 5 \times 7 = 2520$.

The same rule works for three numbers. For 36, 60 and 84:

Only 2 and 3 appear in all three, with smallest powers $2^2$ and $3$, so the HCF is 12. The LCM takes $2^2$, $3^2$, 5 and 7, giving $4 \times 9 \times 5 \times 7 = 1260$. A prime missing from even one number is left out of the HCF, because that number would not be divisible by it.

Coprime numbers

If two numbers are coprime they share no prime, so the HCF is 1 and the LCM is their product. For 8 and 9, $\text{HCF} = 1$ and $\text{LCM} = 72$.

The product of HCF and LCM

For any two positive integers $a$ and $b$,

This follows from the prime-power rule. For each prime, one of the two numbers has the smaller power and the other the larger, so the HCF and LCM together use exactly the powers that $a$ and $b$ use between them. In the example above, $12 \times 2520 = 30240$ and $180 \times 168 = 30240$.

The relation is for two numbers only. For 36, 60 and 84, the HCF times the LCM is $12 \times 1260 = 15120$, while the product of the numbers is $36 \times 60 \times 84 = 181440$, so it does not carry over to three.

The relation is used to find a missing number. If two numbers have HCF 6 and LCM 180 and one of them is 30, the other is

and a check shows $\text{HCF}(30, 36) = 6$ and $\text{LCM}(30, 36) = 180$.

Because the HCF divides both numbers and the LCM is a multiple of both, the HCF must also divide the LCM. A statement such as "two numbers have HCF 8 and LCM 60" therefore cannot be true, since 8 does not divide 60.

Applications

Problems that ask for the greatest size of something that fits several totals exactly are HCF problems, and problems that ask when several repeating events next coincide are LCM problems.

Two bells toll every 18 minutes and every 24 minutes and sound together at noon. They next sound together after the LCM of 18 and 24. With $18 = 2 \times 3^2$ and $24 = 2^3 \times 3$, the LCM is $2^3 \times 3^2 = 72$, so the next time is 72 minutes later.

A hall has 90 chairs in blue and 126 chairs in red, and they are to be set in rows so that each row has chairs of one colour only, all rows have the same number of chairs, and the rows are as long as possible. The row length is the HCF of 90 and 126. Since $90 = 2 \times 3^2 \times 5$ and $126 = 2 \times 3^2 \times 7$, the HCF is $2 \times 9 = 18$, giving $5$ rows of blue and $7$ rows of red.

Using uniqueness

Because a factorisation is unique, it can be used to say what a number can never do. The digit 0 at the end of a whole number means it is divisible by 10, so its factorisation must contain both 2 and 5. A number like $6^n = 2^n \times 3^n$ contains no factor 5 for any natural number $n$, so $6^n$ never ends in 0. By contrast, $20^n = 2^{2n} \times 5^n$ always does.

It can also show that an expression is composite. Take $7 \times 11 \times 13 + 13$. Factoring 13 out gives $13 \times (7 \times 11 + 1) = 13 \times 78$, which has factors other than 1 and itself, so it is composite. In the same way $2 \times 3 \times 5 \times 7 + 7 = 7 \times (2 \times 3 \times 5 + 1) = 7 \times 31$ is composite, even though it looks like the sort of number that might be prime.