In this guide
Number questions are among the fastest in SSC GD maths. They rarely need long calculation. Instead, they test whether you know a handful of rules: what counts as a prime, how place value works, how to test divisibility without dividing, and how unit digits repeat.
Learn these rules once, practise them for a week, and you can answer most number questions in 20–30 seconds. That saves time for the longer arithmetic questions later in the section.
Types of numbers
| Type | Meaning | Examples |
|---|---|---|
| Natural numbers | Counting numbers | 1, 2, 3, 4… |
| Whole numbers | Natural numbers and 0 | 0, 1, 2, 3… |
| Integers | Whole numbers and their negatives | …, −2, −1, 0, 1, 2… |
| Even numbers | Divisible by 2 | 2, 4, 6, 8… |
| Odd numbers | Not divisible by 2 | 1, 3, 5, 7… |
| Prime numbers | Exactly two factors, 1 and itself | 2, 3, 5, 7, 11, 13… |
| Composite numbers | More than two factors | 4, 6, 8, 9, 10… |
The primes below 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Write them out once from memory to check yourself.
Place value and face value
In 45,678:
- the face value of 5 is simply 5;
- the place value of 5 is 5,000, because it sits in the thousands place.
A common question asks for the difference between the two. For 5 in 45,678: 5,000 − 5 = 4,995.
Divisibility rules
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 348 |
| 3 | Sum of digits divisible by 3 | 471 (4 + 7 + 1 = 12) |
| 4 | Last two digits divisible by 4 | 1,316 (16) |
| 5 | Last digit is 0 or 5 | 785 |
| 6 | Divisible by both 2 and 3 | 132 |
| 8 | Last three digits divisible by 8 | 5,120 (120 = 8 × 15) |
| 9 | Sum of digits divisible by 9 | 729 (7 + 2 + 9 = 18) |
| 10 | Last digit is 0 | 450 |
| 11 | Difference between the sums of alternate digits is 0 or a multiple of 11 | 2,871 (2 + 7 = 9; 8 + 1 = 9) |
Combined rules. To test a number like 72 or 12, split it into two numbers that share no common factor and test both. 72 = 8 × 9, 12 = 3 × 4, 15 = 3 × 5. Don't split 12 as 2 × 6: 2 and 6 share the factor 2, so the test fails (for example, 18 passes both but is not divisible by 12).
Smallest and largest multiples
Method for the smallest n-digit multiple: divide the smallest n-digit number by the divisor, then add (divisor − remainder).
Method for the largest n-digit multiple: divide the largest n-digit number by the divisor, then subtract the remainder.
Unit digits of powers
The unit digit of powers repeats in a cycle.
| Base ends in | Cycle of unit digits | Length |
|---|---|---|
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 7 | 7, 9, 3, 1 | 4 |
| 8 | 8, 4, 2, 6 | 4 |
| 4 | 4, 6 | 2 |
| 9 | 9, 1 | 2 |
| 0, 1, 5, 6 | Always the same digit | 1 |
Divide the power by the cycle length. The remainder tells you the position in the cycle. A remainder of 0 means the last position.
Useful sums
- Sum of the first n natural numbers = n(n + 1) ÷ 2. For n = 10: 10 × 11 ÷ 2 = 55.
- Sum of the first n odd numbers = n². For n = 5: 1 + 3 + 5 + 7 + 9 = 25 = 5².
Solved examples
Example 1. Is 7,32,456 divisible by 72?
- 72 = 8 × 9, and 8 and 9 share no common factor.
- Test 8: last three digits 456 = 8 × 57. Yes.
- Test 9: 7 + 3 + 2 + 4 + 5 + 6 = 27, and 27 is divisible by 9. Yes.
- Both pass, so yes. (7,32,456 = 72 × 10,173.)
Example 2. Find the smallest four-digit number divisible by 12.
- The smallest four-digit number is 1,000.
- 1,000 ÷ 12 = 83, remainder 4.
- Add 12 − 4 = 8. Answer: 1,008 (12 × 84 = 1,008).
Example 3. Find the largest four-digit number divisible by 15.
- The largest four-digit number is 9,999.
- 9,999 ÷ 15 = 666, remainder 9 (15 × 666 = 9,990).
- Subtract the remainder: 9,999 − 9 = 9,990.
Example 4. The number 53k6 is divisible by 9, where k is a single digit. Find k.
- Sum of digits = 5 + 3 + k + 6 = 14 + k.
- The next multiple of 9 above 14 is 18, so 14 + k = 18.
- k = 4. (The next option, 27, would need k = 13, which is not a digit.) Check: 5,346 = 9 × 594.
Example 5. Find the unit digit of 7²³.
- The cycle for 7 is 7, 9, 3, 1 (length 4).
- 23 ÷ 4 = 5, remainder 3.
- The third digit in the cycle is 3.
Example 6. A number leaves remainder 4 when divided by 6. What remainder does it leave when divided by 3?
- The number can be written as 6q + 4.
- 6q is divisible by 3, so only 4 matters.
- 4 ÷ 3 leaves remainder 1. Check with 10: 10 ÷ 6 leaves 4, and 10 ÷ 3 leaves 1.
Common mistakes
| Mistake | Correct idea |
|---|---|
| Calling 1 a prime | 1 has only one factor, so it is neither prime nor composite |
| Thinking every odd number is prime | 9, 15, 21 and 27 are odd but composite |
| Testing 12 as 2 × 6 | Use 3 × 4; the two parts must share no factor |
| Remainder 0 in unit-digit cycles taken as "first" | Remainder 0 means the last digit of the cycle |
Practice set
- Which is the smallest prime number?
- Find the difference between the place value and face value of 7 in 37,425.
- Is 5,432 divisible by 8?
- Find the smallest four-digit number divisible by 9.
- Find the largest three-digit number divisible by 7.
- 4k32 is divisible by 11, where k is a single digit. Find k.
- Find the unit digit of 3¹⁰.
- Find the sum of the first 12 odd numbers.
Answers:
- 2.
- 7,000 − 7 = 6,993.
- Last three digits 432 = 8 × 54, so yes.
- 1,000 ÷ 9 = 111, remainder 1; add 8 → 1,008.
- 999 ÷ 7 = 142, remainder 5; 999 − 5 = 994.
- From the right, odd places: 2 + k; even places: 3 + 4 = 7. For a difference of 0, k = 5. Check: 4,532 = 11 × 412.
- Cycle 3, 9, 7, 1; 10 ÷ 4 leaves 2; second digit is 9 (3¹⁰ = 59,049).
- 12² = 144.
What to do next
- Write the divisibility table from memory, then check it.
- Learn the unit-digit cycles for 2, 3, 7 and 8.
- Solve 20 mixed number questions in 10 minutes.
- Move on to LCM and HCF, which builds on factors and primes, and then to squares and square roots.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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