Quantitative Aptitude - MCQ Practice Questions

LCM(9, 11) = 99 (since 9 and 11 are coprime).

Smallest three-digit multiple of 99 is 99 × 2 = 198.

20 = 2² × 5.

Sum of divisors = (1 + 2 + 4)(1 + 5) = 7 × 6 = 42.

Divisors are: 1, 2, 4, 5, 10, 20.

32 ÷ 7 = 4 remainder 4 (No). 33 ÷ 7 = 4 remainder 5 (No). 34 ÷ 7 = 4 remainder 6 (No). 32 = 7×4 + 5 = 28 + 5 (Yes). 32 ÷ 7 gives remainder 5.

The digital root is obtained by repeatedly summing the digits of a number until a single digit remains. Alternatively, the digital root of a positive integer equals the remainder when divided by 9 (or 9 if the remainder is 0).

Step 1: Sum the digits of 9876

Step 2: Sum the digits of 30

Since we've reached a single digit, the process stops.

**Answer: The digital root of 9876 is $3$ (Option D)**

Using HCF × LCM = Product of two numbers: 15 × 180 = 45 × x.

Therefore, 2700 = 45x, so x = 60.

Perfect squares between 100 and 200: 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225 (exceeds 200).

So there are 4 perfect squares.

Wait: 10² = 100 (not between), so squares are 121, 144, 169, 196.

Count = 4.

Actually checking: we need squares from 11² to 14², which is 4 numbers.

To find the remainder when $2^{50}$ is divided by 5, we use the Fermat's Little Theorem approach by finding the pattern of powers of 2 modulo 5.

Step 1: Find the pattern of powers of 2 (mod 5)

Calculate successive powers of 2 and their remainders when divided by 5:

The pattern repeats every 4 powers.

**Step 2: Apply the cycle to find $2^{50}(\mathrm{mod}5)$

Since the remainder repeats with period 4, divide the exponent by 4:

This means $2^{50}$ has the same remainder as $2^2$.

Step 3: Calculate the remainder

**Answer: The remainder is $4$ (Option D)**

We need LCM(12, 15, 20). 12 = 2² × 3, 15 = 3 × 5, 20 = 2² × 5. LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60.

Using HCF × LCM = Product of two numbers: 15 × 180 = 45 × other number.

Therefore, other number = 2700 ÷ 45 = 60.

144 = 2⁴ × 3². 196 = 2² × 7².

Common prime factors: 2².

Therefore, HCF = 4.

Let numbers be 4k and 5k. HCF = k = 6.

Therefore, numbers are 24 and 30.

The larger number is 30.

56 = 2³ × 7, 84 = 2² × 3 × 7, 140 = 2² × 5 × 7.

Common prime factors: 2 × 7 = 14.

16 = 2⁴, 24 = 2³ × 3, 36 = 2² × 3². LCM = 2⁴ × 3² = 16 × 9 = 144.

HCF × LCM = Product of numbers. 18 × 324 = 36 × other.

Therefore, other = 5832 ÷ 36 = 162.

110 = 2 × 5 × 11, 165 = 3 × 5 × 11, 220 = 2² × 5 × 11.

Common factors: 5 × 11 = 55.

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

Therefore, other = 240 ÷ 20 = 12.

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

Other number = 864 ÷ 24 = 36.

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.