LCM / HCF
Q1. Two positive integers a and b satisfy:
- HCF(a, b) = 12
- LCM(a, b) = 180
If a < b, what is the value of b?
A) 60
B) 72
C) 90
D) 108
E) 120
Q2. If the product of two positive integers is 1260 and their GCD is 7, what is their LCM?
A) 180
B) 210
C) 252
D) 270
E) 300
Remainders
Q3. What is the remainder when 4¹⁰⁰ is divided by 7?
A) 1
B) 2
C) 4
D) 5
E) 6
Q4. Find the least positive integer n such that:
n ≡ 1 (mod 2)
n ≡ 2 (mod 3)
n ≡ 3 (mod 4)
n ≡ 4 (mod 5)
A) 59
B) 119
C) 59
D) 39
E) 29
Divisibility Rules
Q5. A five-digit number abcde (each letter represents a digit) is divisible by 11. If a = 9, b = 4, c = 3, and d = 1, what must be the value of e?
A) 5
B) 6
C) 7
D) 4
E) 9
Q6. What is the smallest 4-digit number divisible by both 9 and 12?
A) 1008
B) 1002
C) 1001
D) 1000
E) 1005
Simplification
Q7. Simplify: (1 + 1/2)(1 + 1/3)(1 + 1/4)(1 + 1/5)
A) 2.5
B) 2.8
C) 3
D) 3.2
E) 3.5
Q8. Evaluate: [(25² – 24²)(17² – 16²)] ÷ (49 – 48)
A) 1968
B) 1872
C) 2040
D) 1617
E) 2016
Answer Key and Explanations
A1 A.
Explanation: Use the identity HCF × LCM = a × b. Let a = 12x, b = 12y ⇒ 144xy = 2160 ⇒ xy = 15. Coprime pairs = (3, 5) gives b = 12×5 = 60.
A2 A.
Explanation: Product of two numbers = GCD × LCM ⇒ 1260 = 7 × LCM ⇒ LCM = 180.
A3 C.
Explanation: Powers of 4 modulo 7 repeat in a cycle of 3: (4, 2, 1). 100 ≡ 1 mod 3 ⇒ answer is first in cycle = 4.
A4 A.
Explanation: n ≡ -1 mod 2,3,4,5 ⇒ n + 1 is divisible by LCM(2,3,4,5) = 60 ⇒ n = 59.
A5 D.
Explanation: 9 – 4 + 3 – 1 + e = e + 7 ⇒ divisible by 11 ⇒ e = 4.
A6 A.
Explanation: LCM(9,12) = 36. First 4-digit multiple of 36 after 1000 is 1008.
A7 C.
Explanation: (3/2)(4/3)(5/4)(6/5) = telescopes to 6/2 = 3.
A8 D.
Explanation: (25² – 24²)(17² – 16²)/(49 – 48) = 49 × 33 = 1617.
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