HCF & LCM Word Problems (SSC CGL)

HCF & LCM word problems ka complete guide: kab HCF use karein, kab LCM; bells/alarms, ropes/tiles, grouping, cyclic events, minimum/maximum cases, SSC CGL level solved examples & practice with answers.

---

1) One-line decision rule (sabse important)

2) Type A: Bells / Alarms / Signals (LCM)

Different intervals par event repeat hota hai ⇒ next time together = LCM.

Example 1

Two bells ring every 6 min and 8 min. If they ring together now, after how many minutes will they ring together again?

• Answer = LCM(6,8) = 24 minutes

Example 2 (start time given)

A rings every 12 min, B every 18 min. They rang together at 10:00 AM. When next together?

• LCM(12,18) = 36 min
• Next together at 10:36 AM

3) Type B: Ropes / Wires / Sticks cut into equal pieces (HCF)

“Largest possible equal length pieces” ⇒ HCF.

Example 3

Ropes of length 24 m and 36 m are cut into equal pieces of maximum length. Find length of each piece and number of pieces.

• Max length = HCF(24,36) = 12 m
• Pieces: 24/12 = 2, 36/12 = 3 ⇒ total 5 pieces

4) Type C: Tiles / Square flooring (HCF)

“Largest square tile that fits exactly” ⇒ HCF of dimensions.

Example 4

A rectangular floor is 84 m by 126 m. Find the largest square tile side so that no tile is cut.

• Side = HCF(84,126) = 42 m
• Tiles count = (84/42)×(126/42) = 2×3 = 6

5) Type D: Groups / Distribution / Equal packets (HCF)

“Maximum students per group”, “largest equal packets” ⇒ HCF.

Example 5

90 chocolates and 120 toffees are to be distributed equally among maximum number of children. How many children and how many each?

• Children = HCF(90,120) = 30
• Each gets: 90/30=3 chocolates, 120/30=4 toffees

6) Type E: Meeting / Coincidence / Round trips (LCM)

Different repeat times ⇒ LCM.

Example 6

Three lights blink at intervals 10s, 12s, 15s. If they blink together now, after how many seconds again together?

• LCM(10,12,15) = 60 seconds

7) Mixed clue questions (how to identify)

• Maximum + equal size ⇒ HCF (max common divisor)
• Minimum + together/again ⇒ LCM (min common multiple)
• If “least number that is divisible by all” ⇒ LCM
• If “greatest number that divides all” ⇒ HCF

8) Quick formula reminder

• Two numbers: HCF×LCM = product
• Co-prime: HCF=1 ⇒ LCM = product
• If one divides other: HCF=smaller, LCM=bigger

---

9) Practice (SSC CGL) + Answers

1. Two bells ring at intervals 9 min and 12 min. Together now. Next together after?
2. Find greatest length of a rod that can measure 168 cm and 210 cm exactly.
3. Find least number divisible by 12, 18 and 30.
4. A rectangular park 105 m by 140 m is to be paved with square tiles of max size. Find tile side.
5. Three persons clap every 8s, 10s, 12s. Together now. Next together after?
6. Maximum number of students in rows if 48 boys and 60 girls are to be arranged with equal number in each row and no one left?

Show Answers

1. LCM(9,12)=36 minutes
2. HCF(168,210)=42 cm
3. LCM(12,18,30)=180
4. HCF(105,140)=35 m
5. LCM(8,10,12)=120 seconds
6. HCF(48,60)=12 rows (or groups)