LCM and HCF: AFCAT Numerical Ability

Formulas (3)

• HCF and LCM by prime factors · HCF and LCM from prime powers
  $$\text{HCF} = \prod p^{\min(a, b)}, \qquad \text{LCM} = \prod p^{\max(a, b)}$$
• HCF of large numbers by repeated division · Euclid's division step
  $$\text{HCF}(a, b) = \text{HCF}(b,\; a \bmod b)$$
• HCF and LCM of fractions · Fractions
  $$\text{HCF} = \dfrac{\text{HCF of numerators}}{\text{LCM of denominators}}, \qquad \text{LCM} = \dfrac{\text{LCM of numerators}}{\text{HCF of denominators}}$$

Watch out for (6)

• The HCF keeps only shared primes
  → HCF and LCM by prime factors
• The LCM takes the highest power
  → HCF and LCM by prime factors
• Divide by the remainder
  → HCF of large numbers by repeated division
• The answer is the last divisor
  → HCF of large numbers by repeated division
• The denominators swap jobs
  → HCF and LCM of fractions
• Lowest terms first
  → HCF and LCM of fractions

Formulas (2)

• Product of two numbers equals HCF times LCM · Product rule (two numbers only)
  $$a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$$
• Numbers as the HCF times coprime parts · Coprime parts
  $$a' = Ha,\; b' = Hb,\; \text{HCF}(a, b) = 1 \;\Rightarrow\; \text{LCM} = Hab$$

Reference tables (1)

Watch out for (5)

• Two numbers only
  → Product of two numbers equals HCF times LCM
• Check the answer is a whole number
  → Product of two numbers equals HCF times LCM
• Reduce the ratio first
  → Numbers as the HCF times coprime parts
• The LCM of the ratio terms, not their product
  → Numbers as the HCF times coprime parts
• Ruling out is not proving
  → What the HCF and LCM can and cannot be

Formulas (4)

• The greatest number that divides or measures: tiles, containers and remainders · Greatest divisor leaving remainders
  $$\text{HCF}(a - r_1,\; b - r_2,\; c - r_3)$$
• Bells, lights and laps: when events coincide again · Next time together
  $$\text{next meeting} = \text{start} + \text{LCM}(a, b, c)$$
• The least number leaving given remainders · Least number with a remainder
  $$N = \text{LCM}(a, b, c) \times m + r, \quad m = 1, 2, 3, \ldots$$
• Multiples of the LCM in a range or with a condition · Multiples of the LCM
  $$\text{multiple} = L \times m, \quad L = \text{LCM}(a, b, c)$$

Watch out for (9)

• Take the remainder off first
  → The greatest number that divides or measures: tiles, containers and remainders
• One unit before the HCF
  → The greatest number that divides or measures: tiles, containers and remainders
• Counting the start
  → Bells, lights and laps: when events coincide again
• Together again is an LCM
  → Bells, lights and laps: when events coincide again
• Different remainders, same gap
  → The least number leaving given remainders
• Adding the remainder can add a digit
  → The least number leaving given remainders
• Divisible by all is the LCM, not the product
  → Multiples of the LCM in a range or with a condition
• Largest to subtract leaves one copy, not zero
  → Multiples of the LCM in a range or with a condition
• Read which number is bounded
  → Multiples of the LCM in a range or with a condition