7TH NUMBERS TALENT - LCM - WORD PROBLEMS - 7

Least Common Multiple (LCM) Aptitude Quiz

1. What is the Least Common Multiple (LCM) of 15, 25, and 30?

A) 75 B) 150 C) 225 D) 300

Correct Answer: B) 150

Explanation:
15 = 3 × 5
25 = 5²
30 = 2 × 3 × 5
LCM = 2¹ × 3¹ × 5² = 2 × 3 × 25 = 150.

2. Find the smallest 4-digit number that is completely divisible by 12, 15, and 20.

A) 1000 B) 1020 C) 1080 D) 1200

Correct Answer: B) 1020

Explanation:
LCM(12, 15, 20) = 60.
Smallest 4-digit number = 1000.
1000 ÷ 60 leaves remainder 40.
Required number = 1000 + (60 - 40) = 1020.

3. What is the LCM of two co-prime numbers p and q?

A) 1 B) p + q C) p × q D) p / q

Correct Answer: C) p × q

Explanation: Co-prime numbers share no common factors other than 1 (HCF = 1). Since HCF × LCM = Product of numbers, 1 × LCM = p × q, so LCM = p × q.

4. Calculate the LCM of the fractions 2/3, 4/9, and 5/6.

A) 20/3 B) 10/3 C) 20/9 D) 40/3

Correct Answer: A) 20/3

Explanation: LCM of fractions = LCM of Numerators / HCF of Denominators.
LCM(2, 4, 5) = 20
HCF(3, 9, 6) = 3
Result = 20/3.

5. What is the LCM of (2⁴ × 3² × 5), (2² × 3³ × 7), and (2³ × 3 × 5² × 7²)?

A) 2² × 3 × 5 × 7 B) 2⁴ × 3³ × 5² × 7² C) 2⁴ × 3³ × 5 × 7 D) 2³ × 3² × 5² × 7²

Correct Answer: B) 2⁴ × 3³ × 5² × 7²

Explanation: Select the highest power of each prime factor present in any expression:
Highest power of 2 = 2⁴
Highest power of 3 = 3³
Highest power of 5 = 5²
Highest power of 7 = 7²
LCM = 2⁴ × 3³ × 5² × 7².

6. Three bells toll together at intervals of 9, 12, and 15 minutes respectively. If they toll together now, after how many hours will they toll together next?

A) 2 hours B) 3 hours C) 4 hours D) 5 hours

Correct Answer: B) 3 hours

Explanation: Calculate LCM(9, 12, 15) = 180 minutes.
Convert to hours: 180 ÷ 60 = 3 hours.

7. Rahul and Anita start running around a circular track. Rahul takes 24 seconds per lap, and Anita takes 36 seconds. After how many seconds will they meet at the starting point?

A) 48 seconds B) 72 seconds C) 108 seconds D) 144 seconds

Correct Answer: B) 72 seconds

Explanation: The meeting time at the starting point is given by LCM(24, 36).
24 = 2³ × 3
36 = 2² × 3²
LCM = 2³ × 3² = 8 × 9 = 72 seconds.

8. Traffic lights change every 48, 72, and 108 seconds. If they change simultaneously at 8:00:00 AM, at what time will they change together next?

A) 8:06:12 AM B) 8:07:12 AM C) 8:08:20 AM D) 8:10:00 AM

Correct Answer: B) 8:07:12 AM

Explanation: LCM(48, 72, 108) = 432 seconds.
432 seconds = 7 minutes and 12 seconds.
8:00:00 AM + 7 min 12 sec = 8:07:12 AM.

9. A librarian arranges books in groups of 12, 16, or 18 and always has 5 books left over. What is the minimum total number of books she has?

A) 144 B) 149 C) 139 D) 153

Correct Answer: B) 149

Explanation: Minimum books = LCM(12, 16, 18) + Remainder.
LCM(12, 16, 18) = 144.
144 + 5 = 149 books.

10. A wholesaler packages items into boxes of either 30, 40, or 50 items each without leaving any item unpacked. What is the least total number of items?

A) 300 B) 400 C) 600 D) 1200

Correct Answer: C) 600

Explanation: Total items = LCM(30, 40, 50).
30 = 2 × 3 × 5
40 = 2³ × 5
50 = 2 × 5²
LCM = 2³ × 3 × 5² = 8 × 3 × 25 = 600.

11. Find the smallest number which when divided by 12, 16, 24, and 36 leaves remainders of 7, 11, 19, and 31 respectively.

A) 139 B) 144 C) 135 D) 149

Correct Answer: C) 135

Explanation: Notice common difference (Divisor - Remainder):
12 - 7 = 5, 16 - 11 = 5, 24 - 19 = 5, 36 - 31 = 5.
Required number = LCM(12, 16, 24, 36) - 5.
LCM = 144.
144 - 5 = 135.

12. The HCF of two numbers is 12 and their LCM is 180. If one of the numbers is 36, what is the other number?

A) 45 B) 60 C) 72 D) 90

Correct Answer: B) 60

Explanation: Use formula: HCF × LCM = Product of two numbers.
12 × 180 = 36 × Number₂
Number₂ = (12 × 180) ÷ 36 = 60.

13. What is the greatest 4-digit number that is exactly divisible by 15, 20, and 25?

A) 9850 B) 9900 C) 9930 D) 9960

Correct Answer: B) 9900

Explanation: LCM(15, 20, 25) = 300.
Largest 4-digit number = 9999.
9999 ÷ 300 = 33 with a remainder of 99.
Required number = 9999 - 99 = 9900.

14. Two alarms ring at intervals of 42 minutes and 56 minutes. If both alarms ring together at 12:00 PM, at what time will they ring together again?

A) 2:48 PM B) 3:00 PM C) 2:24 PM D) 3:12 PM

Correct Answer: A) 2:48 PM

Explanation: LCM(42, 56) = 168 minutes.
168 minutes = 2 hours and 48 minutes.
12:00 PM + 2 hrs 48 min = 2:48 PM.

15. Find the sum of digits of the smallest number which when divided by 15, 18, and 24 leaves remainder 8, but is divisible by 13.

A) 17 B) 23 C) 26 D) 29

Correct Answer: B) 23

Explanation: LCM(15, 18, 24) = 360.
The number format is 360k + 8.
Test values of k for divisibility by 13:
For k = 8: 360(8) + 8 = 2888.
2888 ÷ 13 = 222 (exact).
Sum of digits of 2888 = 2 + 8 + 8 + 8 = 23.