Factors of 720

About 720

720 is 1 × 2 × 3 × 4 × 5 × 6, written 6!. It has thirty factors, more than any smaller number. Twelve hours contain 720 minutes.

Factor pairs of 720

A factor pair is two numbers that multiply to give 720: 1 × 720, 2 × 360, 3 × 240, 4 × 180, 5 × 144, 6 × 120, 8 × 90, 9 × 80, 10 × 72, 12 × 60, 15 × 48, 16 × 45, 18 × 40, 20 × 36 and 24 × 30. Each pair is a way of arranging 720 things in equal rows, so 720 tiles can be laid out as a rectangle in 15 different shapes.

How to find the factors of 720 by hand

Divide 720 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 26, because the next number, 27, gives 27 × 27 = 729, which is more than 720: any larger factor has already turned up as the partner of a smaller one.

The numbers that fail are 7, 11, 13, 14, 17, 19, 21, 22, 23, 25 and 26. All the others give a pair.

Prime factorisation of 720

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

Reading down the left side: 720 = 2 × 2 × 2 × 2 × 3 × 3 × 5, which is written 2⁴ × 3² × 5 with powers. 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: (4 + 1) × (2 + 1) × (1 + 1) = 30. This works because a factor of 720 can use each prime anywhere from zero times up to its full power.

The 30 factors add up to 2418. Leaving out 720 itself, the rest add up to 1698, which is more than 720, so 720 is an abundant number. Of its factors, 6 are odd: 1, 3, 5, 9, 15 and 45.

Divisibility rules applied to 720

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 7 + 2 + 0 = 9, which is a multiple of 3.
• By 4: yes. The last two digits make 20, and 20 ÷ 4 = 5.
• 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: yes. 720 ÷ 8 = 90.
• By 9: yes. The digits add up to 9, which is a multiple of 9.
• By 10: yes. The last digit is 0.
• By 11: no. Adding alternate digits from the right gives 7 and 2. Their difference is 5, which is not 0 or a multiple of 11.

HCF and LCM with 720

720 and 625. 720 = 2⁴ × 3² × 5 and 625 = 5⁴. The primes they share, each with its smaller power, give the HCF: 5. Every prime with its larger power gives the LCM: 90000. By division: 720 = 625 × 1 + 95; 625 = 95 × 6 + 55; 95 = 55 × 1 + 40; 55 = 40 × 1 + 15; 40 = 15 × 2 + 10; 15 = 10 × 1 + 5; 10 = 5 × 2 + 0, so the last divisor, 5, is the HCF. Check: 5 × 90000 = 450000 = 720 × 625.

720 and 1000. 720 = 2⁴ × 3² × 5 and 1000 = 2³ × 5³. The primes they share, each with its smaller power, give the HCF: 40. Every prime with its larger power gives the LCM: 18000. By division: 1000 = 720 × 1 + 280; 720 = 280 × 2 + 160; 280 = 160 × 1 + 120; 160 = 120 × 1 + 40; 120 = 40 × 3 + 0, so the last divisor, 40, is the HCF. Check: 40 × 18000 = 720000 = 720 × 1000.

Multiples of 720

Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 720 are 720, 1440, 2160, 2880, 3600, 4320, 5040, 5760, 6480 and 7200. A number has a fixed count of factors but endless multiples.

Common questions

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