Number System made clear.
Recognise the type of number before choosing a shortcut. Divisibility, factors and remainders turn long calculations into small, checkable steps.
Key concepts in Number System
Natural and whole numbers
In this lesson natural numbers start at 1; whole numbers include 0. Integers also include negative whole numbers.
Prime and composite
A prime has exactly two positive factors. 1 is neither prime nor composite; 2 is the only even prime.
Divisibility
For 3 or 9, test the digit sum. For 4, test the last two digits. For 11, the difference between alternating digit sums must be a multiple of 11.
Division algorithm
Dividend = divisor × quotient + remainder, with 0 ≤ remainder < positive divisor.
Factor count
If n = p^a × q^b for distinct primes, its positive factor count is (a+1)(b+1).
Factor tree in numbers
| Stage | Result |
|---|---|
| Start | 72 |
| Split into prime powers | 2³ × 3² |
| Choose exponents | 4 choices × 3 choices |
| Count positive divisors | 12 |
A reliable solving method
- Identify whether the task concerns divisibility, factors, digits or remainders.
- Write a factorisation or division equation rather than guessing.
- Apply a shortcut only after checking its conditions.
- Substitute your result into the original question.
Step-by-step solved examples
Is 4,572 divisible by 9?
- Add the digits: 4+5+7+2 = 18.
- 18 is divisible by 9, so 4,572 is divisible by 9.
- Check: 4,572 ÷ 9 = 508.
How many positive factors does 72 have?
- Factorise: 72 = 2³ × 3².
- Choose the power of 2 in four ways: 0,1,2,3; choose the power of 3 in three ways.
- Total factors = 4 × 3 = 12.
Find the remainder when 7⁴ is divided by 5.
- 7 leaves remainder 2 modulo 5.
- Therefore 7⁴ has the same remainder as 2⁴ = 16.
- 16 = 5 × 3 + 1, so the remainder is 1.
Number System: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
Which is prime?
Show answer and explanation — question 1
Answer C: 29
29 has no positive divisor other than 1 and 29; test primes up to its square root.
Remainder when 125 is divided by 8?
Show answer and explanation — question 2
Answer B: 5
125 = 8 × 15 + 5.
How many positive factors has 36?
Show answer and explanation — question 3
Answer C: 9
36 = 2² × 3², so (2+1)(2+1) = 9.
Which number is divisible by 11?
Show answer and explanation — question 4
Answer A: 121
121 = 11 × 11; its alternating digit-sum difference is 1+1−2 = 0.
Smallest composite positive integer?
Show answer and explanation — question 5
Answer D: 4
4 has factors 1, 2 and 4.
Units digit of 3⁴?
Show answer and explanation — question 6
Answer A: 1
Powers of 3 end in 3,9,7,1, repeating every four powers.
Frequently asked questions
Is zero even?
Yes. It equals 2 × 0, so it is divisible by 2.
Must I list every factor?
No. Prime factorisation gives the count directly; listing is a useful check for small numbers.
Identify whether the task concerns divisibility, factors, digits or remainders. Substitute your result into the original question.