Quantitative Aptitude - MCQ Practice Questions

For two numbers, the fundamental relationship is $\text{HCF}\times \text{LCM}=\text{Product of the two numbers}$. We'll verify which pairs satisfy both the HCF = 7 and LCM = 140 conditions.

Step 1: Apply the HCF-LCM product rule

For any two numbers $a$ and $b$:

Given HCF = 7 and LCM = 140:

Step 2: Check option (A): 7 and 140

Option (A) is valid.

Step 3: Check option (B): 14 and 70

Option (B) is invalid.

Step 4: Check option (C): 28 and 35

Option (C) is valid.

Answer: Both options (A) and (C) satisfy the conditions HCF = 7 and LCM = 140 (Option D)

To find the LCM of three numbers, we use prime factorization and take the highest power of each prime factor.

Step 1: Prime factorization of each number

Step 2: Identify all prime factors and their highest powers

Across all three numbers:

Step 3: Calculate the LCM

Step 4: Final multiplication

**Answer: The LCM of 108, 135, and 162 is $1620$ (Option C)**

96 = 2⁵ × 3, 120 = 2³ × 3 × 5.

Common prime factors: 2³ × 3 = 8 × 3 = 24.

Using HCF × LCM = Product of numbers: 4 × 60 = 20 × other.

Therefore, other = 240 ÷ 20 = 12.

Let numbers be 2k, 3k, 4k. HCF = k = 5.

Numbers are 10, 15, 20.

Sum = 10 + 15 + 20 = 45.

To find the smallest 4-digit number divisible by 12, 18, and 24, we need to find the least common multiple (LCM) and then scale it appropriately.

Step 1: Find the prime factorization of each number

Step 2: Calculate the LCM

For the LCM, take the highest power of each prime factor:

Step 3: Find the smallest 4-digit multiple of 72

The smallest 4-digit number is 1000. Divide it by 72:

Since we need a whole number of groups of 72, round up to the next integer: $14$

Step 4: Calculate the required number

Verify: $1008÷12=84$, $1008÷18=56$, $1008÷24=42$ ✓

**Answer: The smallest 4-digit number divisible by 12, 18, and 24 is $1008$ (Option A)**

Using Euclidean algorithm: 126 = 84 × 1 + 42, then 84 = 42 × 2 + 0.

Therefore, HCF(84, 126) = 42.

Using the formula: HCF × LCM = Product of two numbers. 15 × LCM = 2700, so LCM = 2700 ÷ 15 = 180.

Using HCF × LCM = Product of numbers: 12 × 72 = 24 × other number. 864 = 24 × other number.

Other number = 864 ÷ 24 = 36.

45 = 3² × 5, 60 = 2² × 3 × 5, 75 = 3 × 5².

Common factors: 3 and 5. HCF = 3 × 5 = 15.

20 = 2² × 5, 30 = 2 × 3 × 5, 40 = 2³ × 5. LCM = 2³ × 3 × 5 = 8 × 3 × 5 = 120.

HCF × LCM = Product of numbers. 12 × 144 = 36 × other. 1728 = 36 × other.

Other = 1728 ÷ 36 = 48.

150 = 2 × 3 × 5², 200 = 2³ × 5².

Common factors: 2 and 5². HCF = 2 × 5² = 2 × 25 = 50.

LCM(12, 18) = 36.

Time for both = 36/($\frac{36}{12}$ + $\frac{36}{18}$) = 36/(3 + 2) = $\frac{36}{5}$ = 7.2 minutes.

72 = 2³ × 3², 90 = 2 × 3² × 5.

Common factors: 2 and 3². HCF = 2 × 3² = 2 × 9 = 18.

Let numbers be 3k and 5k.

Their HCF is k = 8.

So numbers are 3(8) = 24 and 5(8) = 40.

12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3. LCM = 2³ × 3² = 8 × 9 = 72.

HCF × LCM = Product of numbers. 36 × LCM = 408 × 468. LCM = (408 × 468) ÷ 36 = 190944 ÷ 36 = 5304.

When three numbers are in a given ratio, we can express them using a common factor. The HCF (Highest Common Factor) is that common factor, and we use it to find the actual numbers.

Step 1: Express the numbers in terms of the common factor

Since the numbers are in ratio $2:3:4$ and their HCF is 5, we can write:

Step 2: Verify the HCF

Check that $\text{HCF}(10,15,20)=5$:

The highest common factor is indeed $5$. ✓

Step 3: Find the LCM

To find $\text{LCM}(10,15,20)$, use prime factorization:

The LCM takes the highest power of each prime:

**Answer: The LCM is $60$ (Option B)**

HCF × LCM = Product of numbers = 360. 360 = 20 × other number.

Other number = 360 ÷ 20 = 18.