FactorsAbout 45
Three quarters of an hour is 45 minutes. 45 is 9 × 5, so its digits add up to 9.
Factor pairs of 45
A factor pair is two numbers that multiply to give 45: 1 × 45, 3 × 15 and 5 × 9. Each pair is a way of arranging 45 things in equal rows, so 45 tiles can be laid out as a rectangle in 3 different shapes.
How to find the factors of 45 by hand
Divide 45 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 6, because the next number, 7, gives 7 × 7 = 49, which is more than 45: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 45? | Result |
|---|---|---|
| 1 | yes | 1 × 45 |
| 2 | no | remainder 1 |
| 3 | yes | 3 × 15 |
| 4 | no | remainder 1 |
| 5 | yes | 5 × 9 |
| 6 | no | remainder 3 |
The numbers that fail are 2, 4 and 6. All the others give a pair.
Prime factorisation of 45
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 45 = 3 × 3 × 5, which is written 32 × 5 with powers. Its prime factors are 3 and 5.
Counting the factors without listing them
Add 1 to each power in the prime factorisation and multiply the results: (2 + 1) × (1 + 1) = 6. This works because a factor of 45 can use each prime anywhere from zero times up to its full power.
The 6 factors add up to 78. Leaving out 45 itself, the rest add up to 33, which is less than 45, so 45 is a deficient number. Of its factors, 6 are odd: 1, 3, 5, 9, 15 and 45.
Divisibility rules applied to 45
The quick tests agree with the factor list above.
- By 2: no. The last digit is 5, which is odd.
- By 3: yes. The digits add up to 4 + 5 = 9, which is a multiple of 3.
- By 4: no. The last two digits make 45, and 45 ÷ 4 leaves remainder 1.
- By 5: yes. The last digit is 5.
- By 6: no. A number divides by 6 when it divides by both 2 and 3. Here the test for 2 fails.
- By 8: no. 45 ÷ 8 leaves remainder 5.
- By 9: yes. The digits add up to 9, which is a multiple of 9.
- By 10: no. The last digit is not 0.
- By 11: no. Adding alternate digits from the right gives 5 and 4. Their difference is 1, which is not 0 or a multiple of 11.
HCF and LCM with 45
45 and 42. 45 = 32 × 5 and 42 = 2 × 3 × 7. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 630. By division: 45 = 42 × 1 + 3; 42 = 3 × 14 + 0, so the last divisor, 3, is the HCF. Check: 3 × 630 = 1890 = 45 × 42.
45 and 48. 45 = 32 × 5 and 48 = 24 × 3. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 720. By division: 48 = 45 × 1 + 3; 45 = 3 × 15 + 0, so the last divisor, 3, is the HCF. Check: 3 × 720 = 2160 = 45 × 48.
Multiples of 45
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 45 are 45, 90, 135, 180, 225, 270, 315, 360, 405 and 450. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 45?
The factors of 45 are 1, 3, 5, 9, 15 and 45. There are 6 of them.
What is the prime factorisation of 45?
45 = 3 × 3 × 5.
Is 45 a perfect square?
No. 6 × 6 = 36 and 7 × 7 = 49, and 45 lies between them.
What is the greatest factor of 45 other than itself?
15. Divide 45 by its smallest prime factor, 3, to get it.