Number System

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Quick recall

• The LCM × HCF identity is for two positive integers.
• The factor-count formula uses the exponents of distinct prime factors.

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Formula Cards

1. LCM and HCF identity

$$LCM(a,b)\,HCF(a,b)=ab$$

For two positive integers.

2. Division algorithm

$$N=dq+r,\qquad0\le r<d$$

For positive integer divisor d; q is the quotient and r the remainder.

3. Divisibility rules

2: last digit even. 3: digit sum divisible by 3. 4: last two digits divisible by 4. 5: ends in 0 or 5. 6: divisible by both 2 and 3. 8: last three digits divisible by 8. 9: digit sum divisible by 9. 11: alternating digit sum divisible by 11, including zero.

4. Number of positive factors

$$N=p_1^{e_1}\cdots p_k^{e_k}\quad\Rightarrow\quad d(N)=\prod_{i=1}^{k}(e_i+1)$$

p₁, …, pₖ are distinct primes. For two prime factors this reduces to (e₁+1)(e₂+1).

Worked Examples

Example 1

Question: Find HCF and LCM of 12 and 18.

Step 1: Prime-factorize both numbers.

$$12=2^2\times3,\qquad18=2\times3^2$$

Step 2: Use lowest powers for HCF and highest for LCM.

$$HCF=2\times3=6,\qquad LCM=2^2\times3^2=36$$

Step 3: Check the product identity.

$$6\times36=12\times18=216$$

Answer: HCF = 6; LCM = 36.

Example 2

Question: Find the remainder when 47 is divided by 6.

Step 1: Find the largest multiple not exceeding 47.

$$6\times7=42$$

Step 2: Subtract.

$$47-42=5$$

Answer: Remainder 5.

Example 3

Question: Is 4536 divisible by 9?

Step 1: Add the digits.

$$4+5+3+6=18$$

Step 2: Check divisibility of the sum.

$$18\div9=2$$

Answer: Yes.

Example 4

Question: Find the number of positive factors of 72.

Step 1: Prime-factorize.

$$72=2^3\times3^2$$

Step 2: Multiply one more than each exponent.

$$(3+1)(2+1)=12$$

Answer: 12 positive factors.

Drill

Try each question before opening its answer.

Question 1

Find LCM and HCF of 15 and 20.

• Show answer

  LCM=60, HCF=5

Question 2

Is 2352 divisible by 8?

• Show answer

  Yes (352÷8=44)

Question 3

Find remainder when 100 is divided by 7.

• Show answer

  2

Question 4

Number of factors of 100.

• Show answer

  9

Question 5

Is 9999 divisible by 11?

• Show answer

  Yes

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