Quantitative Aptitude - MCQ Practice Questions

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)**

First five prime numbers are 2, 3, 5, 7, 11.

Product = 2 × 3 × 5 × 7 × 11 = 2310

This question asks us to identify which number is a perfect square (a number that equals an integer multiplied by itself).

Step 1: Understand Perfect Squares

A perfect square is a number that can be expressed as n × n where n is an integer.

Step 2: Check Each Option

Test each option by finding if its square root is a whole number.

Step 3: Verify the Correct Answer

Only 169 has a whole number square root.

169 is a perfect square because 13 × 13 = 169, making the correct answer (B).

Largest 3-digit number = 999, Smallest 3-digit number = 100.

Difference = 999 - 100 = 899

This question asks us to find an unknown number based on a sequence of arithmetic operations performed on it.

Step 1: Set up the equation

Let the unknown number be x. According to the problem, when x is multiplied by 8 and then 15 is subtracted, the result is 49.

Step 2: Isolate the variable term

Add 15 to both sides of the equation to move the constant to the right side.

Step 3: Solve for the number

Divide both sides by 8 to find the value of x.

The number is 8, which corresponds to answer choice (B).

This question asks us to find the average value of the numbers 1 through 15.

Step 1: Identify the first 15 natural numbers

The first 15 natural numbers are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.

Step 2: Calculate the sum of first 15 natural numbers

Use the formula for sum of first n natural numbers:

Step 3: Calculate the average

Average is the sum divided by the count of numbers.

The average of the first 15 natural numbers is 8.

To find the remainder when a number is divided by another, use the division algorithm: $\text{Dividend}=\text{Divisor}\times \text{Quotient}+\text{Remainder}$

Step 1: Set up the division

We need to express 527 in the form:

where $q$ is the quotient and $r$ is the remainder with $0\le r<15$.

Step 2: Divide 527 by 15

Perform the division:

The quotient is $q=35$.

Step 3: Calculate the product

Step 4: Find the remainder

Subtract the product from the dividend:

Since $0\le 2<15$, this is a valid remainder.

Verification:

To find the remainder when 527 is divided by 15:

15×35=525

Now subtract:

527−525=2

Therefore, the remainder is 2.

**Answer: The remainder is $2$ (Option A)**

Using dividend = divisor × quotient + remainder: Number = 11 × 9 + 5 = 99 + 5 = 104

Let numbers be 5x and 7x.

Then 5x + 7x = 120, so 12x = 120, x = 10.

Larger number = 7 × 10 = 70

Odd numbers between 10 and 30: 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.

Count = 10

Even numbers from 2 to 50: 2, 4, 6, ..., 50.

This is an AP with first term = 2, last term = 50, common difference = 2.

Number of terms = 25.

Sum = 25(2+50)/2 = 25 × 26 = 650