Shortcut Techniques

Problems on H.C.F and L.C.M

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Problems on H.C.F and L.C.M

⚡ Unlock time-saving calculation tricks and mental math techniques. Solve complex problems in seconds with proven shortcut methods used by top performers.

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Shortcut Techniques

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Shortcut Techniques – Problems on H.C.F and L.C.M

Shortcut techniques for H.C.F and L.C.M help candidates solve aptitude questions quickly and accurately in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and placement examinations.

Instead of lengthy calculations, these techniques focus on:

  • Fast factorization methods
  • Quick divisibility checks
  • Prime factor shortcuts
  • Remainder-based tricks
  • Cyclic event calculations
  • Mental calculation techniques
  • Formula-based shortcuts

Why Learn Shortcut Techniques?

  • Reduces solving time significantly.
  • Improves accuracy in competitive exams.
  • Useful in simplification and arithmetic problems.
  • Helps solve cyclic event questions quickly.
  • Improves logical and analytical thinking.

Shortcut #1: Prime Factorization Method

Prime factorization is the fastest method to calculate HCF and LCM of numbers.

Rule for HCF

Take only common prime factors with smallest powers.

Rule for LCM

Take all prime factors with highest powers.

Number Prime Factorization
15 3 × 5
25 5²
27 3³

HCF = 1

LCM = 3³ × 5²

= 675

Shortcut #2: Product Formula

For two numbers:

Product of Two Numbers = HCF × LCM

This shortcut is very useful when one value is missing.

Example:

Two numbers have HCF = 6 and LCM = 180.

Product of numbers = 6 × 180

= 1080

Shortcut #3: HCF and LCM Using Ratios

If ratio of two numbers and HCF are given:

Assume numbers as:

ax and bx

where x = HCF

Example:

Ratio = 3 : 4

HCF = 4

Numbers = 3×4 and 4×4

= 12 and 16

LCM = 48

Shortcut #4: Repeated Event Problems

Whenever events repeat at fixed intervals, use LCM.

Common Applications

  • Bells ringing together
  • Traffic signals
  • Circular track races
  • Machine cycles

Example:

Three bells ring every 6, 8, and 12 minutes.

LCM(6, 8, 12)

= 24

Therefore, bells ring together every 24 minutes.

Shortcut #5: Greatest Number Leaving Same Remainder

If the same remainder is left in each case:

Required Number = HCF of Differences

Example:

Find greatest number dividing 183, 91, and 43 leaving same remainder.

Differences:

183 − 91 = 92

91 − 43 = 48

183 − 43 = 140

HCF(92, 48, 140)

= 4

Shortcut #6: Smallest Number Leaving Same Remainder

When a number leaves same remainder R after division:

Required Number = LCM + Remainder

Example:

Find smallest number leaving remainder 3 when divided by 5, 6, 7, and 8.

LCM(5, 6, 7, 8)

= 840

Required number:

840 + 3

= 843

Shortcut #7: HCF of Fractions

Formula:

HCF of Fractions = HCF of Numerators / LCM of Denominators

Example:

Find HCF of 2/3 and 4/9

HCF of numerators = 2

LCM of denominators = 9

HCF = 2/9

Shortcut #8: LCM of Fractions

Formula:

LCM of Fractions = LCM of Numerators / HCF of Denominators

Example:

Find LCM of 2/3 and 4/9

LCM of numerators = 4

HCF of denominators = 3

LCM = 4/3

Shortcut #9: Fast LCM Calculation Using Division Table

For large numbers, use common division method instead of listing multiples.

Division Numbers
2 12, 18, 24
2 6, 9, 12
3 3, 9, 6
2 1, 3, 2
3 1, 3, 1

LCM = 2 × 2 × 3 × 2 × 3

= 72

Shortcut #10: Divisibility Shortcut

Before solving HCF or LCM problems, check divisibility rules quickly.

Number Shortcut Rule
2 Last digit even
3 Sum of digits divisible by 3
5 Last digit 0 or 5
9 Sum of digits divisible by 9
11 Difference of alternate digit sums divisible by 11

Shortcut #11: Circular Track Problems

For runners moving around a circular track:

Use LCM of individual completion times.

Example:

A completes one round in 252 sec.

B completes one round in 308 sec.

C completes one round in 198 sec.

LCM(252, 308, 198)

= 2772 sec

= 46 min 12 sec

Shortcut #12: Elimination Technique in MCQs

Use option elimination using:

  • Divisibility rules
  • Even/Odd properties
  • Prime factor patterns
  • Approximation methods

✔ Eliminating wrong options saves valuable exam time.

Important Exam Tips

  • Memorize prime numbers up to 50.
  • Practice factorization regularly.
  • Use HCF for greatest divisor problems.
  • Use LCM for repeated event problems.
  • Remember product relation: HCF × LCM.
  • Use divisibility tests before calculation.
  • Avoid lengthy multiplication whenever possible.

Shortcut techniques in H.C.F and L.C.M help candidates improve speed, reduce calculation errors, and solve aptitude questions efficiently in competitive examinations.

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