Quantitative Aptitude - MCQ Practice Questions

Step 1: From first investment, 2800 = (P × 14 × 5) / 100 = 0.7P.

Step 2: P = $\frac{2800}{0}$.7 = ₹4000.

Step 3: At 10% for 5 years, SI = (4000 × 10 × 5) / 100 = $\frac{200000}{100}$ = ₹2000.

Step 1: If amount doubles, SI = P.

So P = (P × R × 8) / 100, giving R = $\frac{100}{8}$ = 12.5% per annum.

Step 2: For amount to become 3 times, SI = 2P.

Step 3: 2P = (P × 12.5 × T) / 100, so 2 = 0.125T, therefore T = $\frac{2}{0}$.125 = 16 years.

Sum of first n natural numbers = n(n+1)/2.

Here n=10, so sum = 10(11)/2 = $\frac{110}{2}$ = 55

Check: 21=3×7 (not prime), 23 is only divisible by 1 and 23 (prime), 25=5×5 (not prime), 27=3×9 (not prime).

So 23 is the smallest prime greater than 20.

Factors of 48: 1,2,3,4,6,8,12,16,24,48.

Factors of 64: 1,2,4,8,16,32,64.

Common factors: 1,2,4,8,16. HCF = 16

If a number is divisible by both 3 and 5, and 3 and 5 are coprime (HCF=1), then the number must be divisible by their product: 3×5 = 15

This question asks us to find the smallest positive number that is divisible by both 12 and 18.

Step 1: Find Prime Factorization of Both Numbers

Break down each number into its prime factors.

Step 2: Identify Highest Powers of All Prime Factors

The LCM uses the highest power of each prime factor that appears.

Step 3: Calculate the LCM

Multiply the highest powers together.

The LCM of 12 and 18 is 36, which is the smallest number divisible by both numbers.

Numbers of form 7k+3: when k=5, number=7(5)+3=38.

Check: 38÷7=5 remainder 3 ✓.

Check others: 24÷7=3 rem 3 (close but let's verify 38 first), 38÷7 gives remainder 3 ✓

Let smaller odd number = x.

Then x+(x+2)=56.

So 2x+2=56, 2x=54, x=27.

Check: 27+29=56 ✓

A number has exactly 3 factors only when it is the square of a prime.

For p² where p is prime, factors are: 1, p, p².

Example: 4 has factors 1,2,4 (3 factors). 9 has factors 1,3,9 (3 factors).

Divisors of 28: 1, 2, 4, 7, 14, 28.

Sum = 1+2+4+7+14+28 = 56. (Note: 28 is a perfect number where sum of proper divisors = 28)

When a number is expressed in prime factorization form, we use the divisor formula: if $n=p_1^{a_1}\times p_2^{a_2}\times p_3^{a_3}$, then the total number of divisors is $(a_1+1)(a_2+1)(a_3+1)$.

Step 1: Identify the prime factorization

The given number is:

Here, the exponents are: $a_1=3$, $a_2=2$, $a_3=1$

Step 2: Apply the divisor formula

The number of divisors is found by adding 1 to each exponent and multiplying:

Step 3: Calculate

Step 4: Verify with a divisor example

Each divisor has the form $2^a\times 3^b\times 5^c$ where $0\le a\le 3$, $0\le b\le 2$, $0\le c\le 1$. This gives 4 choices for the power of 2, 3 choices for the power of 3, and 2 choices for the power of 5.

**Answer: The total number of divisors is $24$ (Option B)**

Using prime factorization: 48 = 2⁴ × 3, 64 = 2⁶. HCF = 2⁴ = 16 (taking lowest powers of common prime factors).

Prime factorization: 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 4 × 9 = 36 (taking highest powers of all prime factors).

This question asks us to find the Highest Common Factor (HCF) of two numbers using prime factorization or the Euclidean algorithm.

Step 1: Prime factorization of 56

Express 56 as a product of prime numbers.

Step 2: Prime factorization of 72

Express 72 as a product of prime numbers.

Step 3: Find the HCF

The HCF is the product of common prime factors with their lowest powers.

The HCF of 56 and 72 is 8, making the correct answer (B).

This question asks us to find the Least Common Multiple (LCM) of two numbers using prime factorization.

Step 1: Find prime factorization of 15

Break 15 into its prime factors.

Step 2: Find prime factorization of 25

Break 25 into its prime factors.

Step 3: Calculate LCM using highest powers of all prime factors

The LCM is found by taking the highest power of each prime that appears in either factorization: 3¹ and 5².

The LCM of 15 and 25 is 75.

36 = 2² × 3², 48 = 2⁴ × 3. HCF = 2² × 3 = 12, LCM = 2⁴ × 3² = 144.

Product = 36 × 48 = HCF × LCM = 12 × 144 = 1728.

Using formula: HCF × LCM = Product of two numbers. 8 × 96 = 24 × x. 768 = 24x. x = 32.

100 = 2² × 5², 150 = 2 × 3 × 5², 200 = 2³ × 5².

Common factors: 2¹ × 5² = 2 × 25 = 50.

This question asks us to find the least common multiple (LCM) of two numbers using prime factorization.

Step 1: Find prime factorization of 84

Break 84 into its prime factors by dividing by smallest primes.

Step 2: Find prime factorization of 140

Break 140 into its prime factors by dividing by smallest primes.

Step 3: Calculate LCM using highest powers of all prime factors

LCM is found by taking the highest power of each prime factor that appears in either number.

The LCM of 84 and 140 is 420, which is option (A).