Definition

Coprime means no shared prime factor

Arithmetic • Number System • LCM and HCF

Statement

Two positive integers

p

and

q

are coprime when

HCF (p, q) = 1

, which holds exactly when their prime factorisations have no prime in common; neither number need itself be prime.

How to use it

1To test a pair, list the prime factors of each and check the two lists are disjoint.
2To show a described case is possible, produce one witness: $2$ and $3$ , $3$ and $4$ , $4$ and $9$ cover prime-prime, prime-composite and composite-composite.
3If $HCF (p, q) = 1$ then $LCM (p, q) = pq$ .

Are

35

and

54

coprime?

Step-by-step

1
$35 = 5 \times 7$ and $54 = 2 \times 3^{3}$
Why: Factorise each number into primes.
2
${5, 7} \cap {2, 3} = \emptyset$ , so $HCF (35, 54) = 1$

Answer

Yes, they are coprime

Watch out

Coprimality is a property of the pair, not of either number:

9

and

16

are coprime although both are composite, while

6

and

9

are not although neither is prime. For three or more numbers, pairwise coprime and jointly coprime are different conditions.

Builds on

Also known as

coprime, relatively prime, mutually prime, HCF equals 1, gcd 1