Number System

Most number questions never need the full number. Divisibility wants the last few digits, unit digit wants the last digit, remainders want small leftovers. Throw away everything the question does not ask about.

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Most number questions never need the whole number. 'Is it divisible by 8?' needs the last 3 digits. 'What is the unit digit?' needs the last digit. Learn which part to look at.

Number system is the most common quant topic in placement papers: divisibility, unit digit, remainders, factors, HCF and LCM. Each one has a rule that avoids big calculations, and those rules are also used inside every other chapter.

Kinds of numbers

Natural numbers: 1, 2, 3 ... Whole numbers: 0, 1, 2, 3 ... Integers add the negatives: ... -2, -1, 0, 1, 2 ...

A prime has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ... 1 is NOT prime, and 2 is the only even prime. A composite number has more than two factors (4, 6, 8, 9 ...).

To test if a number up to 400 is prime, divide only by primes up to its square root. For 221: √(221) is about 14.9, so try 2, 3, 5, 7, 11, 13. 13 × 17 = 221, so it is not prime.

Co-primes are two numbers whose only common factor is 1, like 8 and 15. They do not have to be prime themselves.

Kinds of numbers: reference

Worked example

Is 391 a prime number?

1. √(391) is just under 20, so test primes up to 19.
   Why: If 391 had a factor bigger than 20, it would also have one smaller than 20.
2. 2, 3, 5, 7, 11, 13: none divides. 17 × 23 = 391.
   Why: Check each prime in turn.
3. Not prime: 391 = 17 × 23.

Answer: Not prime

Divisibility rules

You can tell whether a number divides another without dividing. Each rule looks at only part of the number.

2, 5, 10: look at the last digit. 4: last two digits. 8: last three digits. 3 and 9: add all the digits. 11: alternate plus and minus across the digits; the result must be 0 or a multiple of 11.

For a composite divisor, split it into co-prime parts: divisible by 12 means divisible by 3 AND 4; by 72 means by 8 AND 9.

Divisibility rules: reference

Worked example

Find the smallest digit * so that 5*2 is divisible by 6.

1. Divisible by 6 means divisible by 2 and by 3.
   Why: 6 = 2 × 3, and 2 and 3 are co-prime.
2. It ends in 2, so it is already even.
3. Digit sum 5 + * + 2 = 7 + * must be divisible by 3, so * = 2, 5 or 8.
4. Smallest is 2.

Answer: 2

Worked example

Is 7,29,432 divisible by 72?

1. 72 = 8 × 9, co-prime parts.
   Why: Check both rules; both must pass.
2. Last three digits 432 / 8 = 54, yes.
   Why: Rule for 8.
3. Digit sum 7+2+9+4+3+2 = 27, divisible by 9, yes.
   Why: Rule for 9.
4. Yes.

Answer: Yes

Unit digit of big powers

Worked example

Unit digit of 7⁹⁵.

1. 7's cycle: 7, 9, 3, 1 (length 4).
   Why: 7¹ = 7, 7² = 49, 7³ = 343, 7⁴ = 2401, then it repeats.
2. 95 = 4 × 23 + 3, remainder 3.
   Why: Full cycles do not change the position.
3. Third in the cycle: 3.

Answer: 3

Worked example

Unit digit of 2⁵³ × 3⁶².

1. 53 = 4 × 13 + 1, so 2⁵³ ends like 2¹ = 2.
2. 62 = 4 × 15 + 2, so 3⁶² ends like 3² = 9.
3. 2 × 9 = 18, unit digit 8.

Answer: 8

Exam tip

Factors, and how many a number has

Worked example

How many factors does 720 have?

1. 720 = 2⁴ × 3² × 5.
   Why: Prime factorisation.
2. (4+1)(2+1)(1+1) = 5 × 3 × 2 = 30.
   Why: Add one to each power, multiply.

Answer: 30

HCF and LCM

HCF (highest common factor) is the biggest number that divides all of them. LCM (lowest common multiple) is the smallest number all of them divide.

From prime factors: HCF takes the LOWEST power of each common prime; LCM takes the HIGHEST power of every prime present.

For two numbers only: HCF × LCM = product of the numbers.

Word clues: 'largest', 'greatest', 'maximum size of tiles' → HCF. 'Smallest', 'least', 'when will they meet again', 'bells ring together' → LCM.

HCF and LCM: the rules

Worked example

Three bells ring every 12, 15 and 20 minutes. They ring together at 9:00. When do they next ring together?

1. They ring together at common multiples of 12, 15 and 20.
   Why: 'Together again' is an LCM clue.
2. LCM(12, 15, 20) = 60 minutes.
   Why: 12 = 2² × 3, 15 = 3 × 5, 20 = 2² × 5 → 2² × 3 × 5.
3. At 10:00.

Answer: 60 minutes

Worked example

Find the greatest number that divides 43, 91 and 183 leaving the same remainder.

1. Differences: 91 - 43 = 48, 183 - 91 = 92.
   Why: If both leave remainder r, the remainder cancels in the difference.
2. HCF(48, 92) = 4.
3. 4.

Answer: 4

Remainders

The remainder of a sum or product can be found from the remainders of the parts. 17 × 23 / 12: 17 leaves 5 and 23 leaves 11, 5 × 11 = 55, which leaves 7.

For big powers, find a small power that leaves remainder 1, then use it. 2³ = 8 leaves 1 when divided by 7, so 2⁹⁹ = (2³)³³ leaves 1 too.

Negative remainders are allowed and often faster: 23 leaves -1 when divided by 12.

Worked example

Remainder when 2¹⁰⁰ is divided by 7.

1. 2³ = 8 = 7 + 1 leaves remainder 1.
   Why: Look for a power just above a multiple of 7.
2. 2¹⁰⁰ = (2³)³³ × 2.
   Why: 100 = 3 × 33 + 1.
3. Remainder = 1³³ × 2 = 2.
   Why: Each 2³ contributes a remainder of 1.

Answer: 2

Worked example

Remainder when 25²⁵ is divided by 26.

1. 25 = 26 - 1 leaves remainder -1.
   Why: A negative remainder is often easier.
2. (-1)²⁵ = -1.
   Why: Odd power keeps the minus sign.
3. -1 means 26 - 1 = 25.

Answer: 25

Trailing zeros of n!

A zero at the end comes from a factor 10 = 2 × 5. In n! there are always more 2s than 5s, so count the 5s.

Zeros in n! = n/5 + n/25 + n/125 + ... (drop the remainders each time).

Worked example

How many zeros does 100! end with?

1. 100/5 = 20.
   Why: Every multiple of 5 gives one 5.
2. 100/25 = 4.
   Why: Multiples of 25 give a second 5.
3. 20 + 4 = 24.

Answer: 24

Shortcut: Greatest / smallest fraction without decimals

Use it when: Four fractions in the options.

1. Cross-multiply two at a time: a/b vs c/d → compare a*d with b*c.
2. If numerators rise by the same step and denominators by the same step, compare that step fraction to the first: if step fraction is bigger, the list is rising.

Greatest / smallest fraction without decimals: try it

Largest of 3/5, 5/8, 7/11, 9/14

1. Numerators +2, denominators +3. Step fraction 2/3 = 0.67.
2. 2/3 is bigger than 3/5, so each next fraction is bigger. Last one wins.

Answer: 9/14

Shortcut: Remainder of a huge power

Use it when: 2¹⁰⁰ / 7 style.

1. Find the smallest power leaving remainder 1 (2³ = 8 → remainder 1).
2. Write the exponent as multiples of that plus leftover: 2¹⁰⁰ = (2³)³³ × 2.
3. (1)³³ × 2 = 2.

Remainder of a huge power: try it

Remainder of 2¹⁰⁰ / 7

1. 2³ leaves 1.
2. 2⁹⁹ leaves 1, times 2.

Answer: 2

Common mistakes

Common mistakes

Your turn medium

Unit digit of 7⁹⁵ × 3⁵⁸

1. 95 mod 4 = 3 → 7's cycle → 3.
2. 58 mod 4 = 2 → 3's cycle → 9.
3. 3 × 9 = 27 → 7.

Answer: 7

Your turn medium

Remainder when 17 × 23 × 29 is divided by 12

1. 17 → 5, 23 → 11, 29 → 5.
2. 5 × 11 = 55 → 7. 7 × 5 = 35 → 11.

Answer: 11

Your turn medium

Largest 4-digit number divisible by 12, 15 and 18

1. LCM = 180.
2. 9999 / 180 = 55 rem 99.
3. 9999 - 99 = 9900.

Answer: 9900

Your turn hard

Greatest number that divides 1657 and 2037 leaving remainders 6 and 5

1. Subtract the remainders: 1651 and 2032.
2. HCF(1651, 2032) = 127.

Answer: 127

Recap

• test a number for 3, 4, 8, 9 and 11 without dividing
• find the unit digit of a big power from its cycle
• count the factors of a number from its prime powers
• choose HCF or LCM from the words in the question
• find the remainder of a big power

Now practise

19 questions with full solutions.

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