Factors of 30

About 30

30 is 2 × 3 × 5, the product of the first three primes. It is the smallest number with three different prime factors.

Factor pairs of 30

A factor pair is two numbers that multiply to give 30: 1 × 30, 2 × 15, 3 × 10 and 5 × 6. Each pair is a way of arranging 30 things in equal rows, so 30 tiles can be laid out as a rectangle in 4 different shapes.

How to find the factors of 30 by hand

Divide 30 by 1, 2, 3 and so on. Each time the division is exact, you have found two factors at once: the divisor and the answer. You can stop at 5, because the next number, 6, gives 6 × 6 = 36, which is more than 30: any larger factor has already turned up as the partner of a smaller one.

The only number that fails is 4. All the others give a pair.

Prime factorisation of 30

Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.

Reading down the left side: 30 = 2 × 3 × 5. Its prime factors are 2, 3 and 5.

Counting the factors without listing them

Add 1 to each power in the prime factorisation and multiply the results: (1 + 1) × (1 + 1) × (1 + 1) = 8. This works because a factor of 30 can use each prime anywhere from zero times up to its full power.

The 8 factors add up to 72. Leaving out 30 itself, the rest add up to 42, which is more than 30, so 30 is an abundant number. Of its factors, 4 are odd: 1, 3, 5 and 15.

Divisibility rules applied to 30

The quick tests agree with the factor list above.

• By 2: yes. The last digit is 0, which is even.
• By 3: yes. The digits add up to 3 + 0 = 3, which is a multiple of 3.
• By 4: no. The last two digits make 30, and 30 ÷ 4 leaves remainder 2.
• By 5: yes. The last digit is 0.
• By 6: yes. A number divides by 6 when it divides by both 2 and 3. Here both tests pass.
• By 8: no. 30 ÷ 8 leaves remainder 6.
• By 9: no. The digits add up to 3, which is not a multiple of 9.
• By 10: yes. The last digit is 0.
• By 11: no. Adding alternate digits from the right gives 0 and 3. Their difference is 3, which is not 0 or a multiple of 11.

HCF and LCM with 30

30 and 28. 30 = 2 × 3 × 5 and 28 = 2² × 7. The primes they share, each with its smaller power, give the HCF: 2. Every prime with its larger power gives the LCM: 420. By division: 30 = 28 × 1 + 2; 28 = 2 × 14 + 0, so the last divisor, 2, is the HCF. Check: 2 × 420 = 840 = 30 × 28.

30 and 32. 30 = 2 × 3 × 5 and 32 = 2⁵. The primes they share, each with its smaller power, give the HCF: 2. Every prime with its larger power gives the LCM: 480. By division: 32 = 30 × 1 + 2; 30 = 2 × 15 + 0, so the last divisor, 2, is the HCF. Check: 2 × 480 = 960 = 30 × 32.

Multiples of 30

Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270 and 300. A number has a fixed count of factors but endless multiples.

Common questions

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