FactorsAbout 75
Three quarters of 100. 75 is 3 × 25.
Factor pairs of 75
A factor pair is two numbers that multiply to give 75: 1 × 75, 3 × 25 and 5 × 15. Each pair is a way of arranging 75 things in equal rows, so 75 tiles can be laid out as a rectangle in 3 different shapes.
How to find the factors of 75 by hand
Divide 75 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 8, because the next number, 9, gives 9 × 9 = 81, which is more than 75: any larger factor has already turned up as the partner of a smaller one.
| Try | Divides 75? | Result |
|---|---|---|
| 1 | yes | 1 × 75 |
| 2 | no | remainder 1 |
| 3 | yes | 3 × 25 |
| 4 | no | remainder 3 |
| 5 | yes | 5 × 15 |
| 6 | no | remainder 3 |
| 7 | no | remainder 5 |
| 8 | no | remainder 3 |
The numbers that fail are 2, 4, 6, 7 and 8. All the others give a pair.
Prime factorisation of 75
Divide by the smallest prime that goes in exactly, and repeat with the answer until you reach 1.
Reading down the left side: 75 = 3 × 5 × 5, which is written 3 × 52 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: (1 + 1) × (2 + 1) = 6. This works because a factor of 75 can use each prime anywhere from zero times up to its full power.
The 6 factors add up to 124. Leaving out 75 itself, the rest add up to 49, which is less than 75, so 75 is a deficient number. Of its factors, 6 are odd: 1, 3, 5, 15, 25 and 75.
Divisibility rules applied to 75
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 7 + 5 = 12, which is a multiple of 3.
- By 4: no. The last two digits make 75, and 75 ÷ 4 leaves remainder 3.
- 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. 75 ÷ 8 leaves remainder 3.
- 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 5 and 7. Their difference is 2, which is not 0 or a multiple of 11.
HCF and LCM with 75
75 and 72. 75 = 3 × 52 and 72 = 23 × 32. The primes they share, each with its smaller power, give the HCF: 3. Every prime with its larger power gives the LCM: 1800. By division: 75 = 72 × 1 + 3; 72 = 3 × 24 + 0, so the last divisor, 3, is the HCF. Check: 3 × 1800 = 5400 = 75 × 72.
75 and 80. 75 = 3 × 52 and 80 = 24 × 5. The primes they share, each with its smaller power, give the HCF: 5. Every prime with its larger power gives the LCM: 1200. By division: 80 = 75 × 1 + 5; 75 = 5 × 15 + 0, so the last divisor, 5, is the HCF. Check: 5 × 1200 = 6000 = 75 × 80.
Multiples of 75
Factors divide a number. Multiples are what you get by multiplying it. The first ten multiples of 75 are 75, 150, 225, 300, 375, 450, 525, 600, 675 and 750. A number has a fixed count of factors but endless multiples.
Common questions
What are the factors of 75?
The factors of 75 are 1, 3, 5, 15, 25 and 75. There are 6 of them.
What is the prime factorisation of 75?
75 = 3 × 5 × 5.
Is 75 a perfect square?
No. 8 × 8 = 64 and 9 × 9 = 81, and 75 lies between them.
What is the greatest factor of 75 other than itself?
25. Divide 75 by its smallest prime factor, 3, to get it.