HCF & LCM
🔒 Log in to trackHighest common factor and lowest common multiple — a small but near-guaranteed slot in CGL. Most questions are template word problems: greatest divisor with remainders (HCF of differences), least number with given remainders (LCM + r), bells, tiles and ratio pairs.
One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.
Every formula on one printable page, grouped by subtopic.
5 exam-level questions worked step by step.
56 questions — untimed practice or a timed test with analysis.
Track record in the exam
Questions per shift in recent SSC CGL papers.
Test difficulty mix (56 questions)
Question patterns exams keep repeating
Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.
Greatest divisor with given remainders
very common"Greatest number that divides 82, 150 and 233 leaving remainders 4, 7 and 12."
Subtract each remainder from its number.
Take the HCF of the answers.
Find the greatest number which divides 82, 150 and 233 leaving remainders 4, 7 and 12 respectively.
82 − 4 = 78, 150 − 7 = 143, 233 − 12 = 221
These are 13 × 6, 13 × 11 and 13 × 17
6, 11 and 17 share no common factor
HCF = 13
Least number with the same remainder
very common"Least number which when divided by 8, 12, 15, 20 leaves remainder 4 each time."
Find the LCM of the divisors.
Add the remainder.
Find the least number which when divided by 8, 12, 15 and 20 leaves remainder 4 in each case.
LCM of 8, 12, 15, 20 = 120
120 divides by all four with nothing left over
Add the remainder: 120 + 4 = 124
Remainder is always the same amount short
commonRemainders differ but each is the same amount less than its divisor (e.g. 5, 8, 11 for 6, 9, 12).
Check that divisor − remainder is the same each time.
Answer = LCM − that amount.
Find the least number which when divided by 6, 9 and 12 leaves remainders 5, 8 and 11 respectively.
6 − 5 = 1, 9 − 8 = 1, 12 − 11 = 1 (always 1 short)
LCM of 6, 9, 12 = 36
36 − 1 = 35
Ratio with HCF or LCM, find the numbers
very common"Two numbers are in the ratio 5 : 7 and their LCM is 140."
Call the numbers 5h and 7h (h is the HCF).
LCM = 5 × 7 × h, so find h.
Two numbers are in the ratio 5 : 7 and their LCM is 140. Find the numbers.
Numbers: 5h and 7h
LCM = 5 × 7 × h = 35h
35h = 140, so h = 4
Numbers: 5 × 4 and 7 × 4 = 20 and 28
Product = HCF × LCM
commonTwo of these are given: product, HCF, LCM, one number. Find the other.
For two numbers: HCF × LCM = product of the numbers.
Divide to find the missing one.
The product of two numbers is 2160 and their HCF is 12. Find their LCM.
HCF × LCM = product
12 × LCM = 2160
LCM = 2160 ÷ 12 = 180
Bells ringing together again
commonThings repeat at fixed gaps and start together; asks when they next meet.
Find the LCM of the gaps (same unit).
Add it to the start time.
Three bells toll at intervals of 9, 12 and 15 minutes, together at 8 a.m. When do they next toll together?
9 = 3 × 3, 12 = 2 × 2 × 3, 15 = 3 × 5
LCM = 2 × 2 × 3 × 3 × 5 = 180 minutes = 3 hours
8 a.m. + 3 hours = 11 a.m.
Largest tile, rod or vessel
common"Largest square tile", "longest equal pieces" or "biggest vessel that measures exactly".
Make all units the same.
Take the HCF.
What is the largest vessel that can fill 144 L, 180 L and 240 L containers exactly?
144 = 12 × 12, 180 = 12 × 15, 240 = 12 × 20
12, 15 and 20 share no common factor
Largest vessel = 12 L
Least or greatest n-digit number
common"Least five-digit number exactly divisible by …" or "greatest four-digit number …".
Find the LCM.
Least: go up from 10000 to the next multiple. Greatest: go down from 9999.
Find the least five-digit number exactly divisible by 32, 36, 40, 45 and 48.
LCM = 1440
1440 × 6 = 8640 (only four digits)
1440 × 7 = 10080 (first five-digit multiple)
Answer = 10080
HCF of decimals or fractions
occasionalA list like 0.54, 1.8, 7.2, or fractions like 2/3, 8/9, 10/27.
Decimals: multiply all by 10 or 100 to remove the point, find the HCF, then put the point back.
Fractions: HCF = HCF of tops ÷ LCM of bottoms.
Find the HCF of 0.54, 1.8 and 7.2.
Multiply each by 100: 54, 180, 720
HCF of 54, 180, 720 = 18
Divide by 100 again: 18 ÷ 100 = 0.18