FactorsAbout 84
84 is 12 × 7: the number of months in 7 years and of days in 12 weeks.
Factor pairs of 84
A factor pair is two numbers that multiply to give 84: 1 × 84, 2 × 42, 3 × 28, 4 × 21, 6 × 14 and 7 × 12. Each pair is a way of arranging 84 things in equal rows, so 84 tiles can be laid out as a rectangle in 6 different shapes.
How to find the factors of 84 by hand
Divide 84 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 9, because the next number, 10, gives 10 × 10 = 100, which is more than 84: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 84? | Result |
|---|---|---|
| 1 | yes | 1 × 84 |
| 2 | yes | 2 × 42 |
| 3 | yes | 3 × 28 |
| 4 | yes | 4 × 21 |
| 5 | no | remainder 4 |
| 6 | yes | 6 × 14 |
| 7 | yes | 7 × 12 |
| 8 | no | remainder 4 |
| 9 | no | remainder 3 |
The numbers that fail are 5, 8 and 9. All the others give a pair.
Prime factorisation of 84
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 84 = 2 × 2 × 3 × 7, which is written 22 × 3 × 7 with powers. Its prime factors are 2, 3 and 7.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (2 + 1) × (1 + 1) × (1 + 1) = 12. This works because a factor of 84 can use each prime anywhere from zero times up to its full power.
The 12 factors add up to 224. Leaving out 84 itself, the rest add up to 140, which is more than 84, so 84 is an abundant number. Of its factors, 4 are odd: 1, 3, 7 and 21.
Divisibility rules applied to 84
The quick tests agree with the factor list above.
- By 2: yes. The last digit is 4, which is even.
- By 3: yes. The digits add up to 8 + 4 = 12, which is a multiple of 3.
- By 4: yes. The last two digits make 84, and 84 ÷ 4 = 21.
- By 5: no. The last digit is 4, not 0 or 5.
- By 6: yes. A number divides by 6 when it divides by both 2 and 3. Here both tests pass.
- By 8: no. 84 ÷ 8 leaves remainder 4.
- By 9: no. The digits add up to 12, which is not a multiple of 9.
- By 10: no. The last digit is not 0.
- By 11: no. Adding alternate digits from the right gives 4 and 8. Their difference is 4, which is not 0 or a multiple of 11.
HCF and LCM with 84
84 and 81. 84 = 22 × 3 × 7 and 81 = 34. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 2268. By division: 84 = 81 × 1 + 3; 81 = 3 × 27 + 0, so the last divisor, 3, is the HCF. Check: 3 × 2268 = 6804 = 84 × 81.
84 and 90. 84 = 22 × 3 × 7 and 90 = 2 × 32 × 5. The primes they share, each with its smaller power, give the HCF: 6. Every prime with its larger power gives the LCM: 1260. By division: 90 = 84 × 1 + 6; 84 = 6 × 14 + 0, so the last divisor, 6, is the HCF. Check: 6 × 1260 = 7560 = 84 × 90.
Multiples of 84
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 84 are 84, 168, 252, 336, 420, 504, 588, 672, 756 and 840. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 84?
The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. There are 12 of them.
What is the prime factorisation of 84?
84 = 2 × 2 × 3 × 7.
Is 84 a perfect square?
No. 9 × 9 = 81 and 10 × 10 = 100, and 84 lies between them.
What is the greatest factor of 84 other than itself?
42. Divide 84 by its smallest prime factor, 2, to get it.