Numbers Aptitude

Numbers form the basis of mathematics and are integral in various quantitative aptitude tests. These questions often involve concepts like divisibility, prime numbers, odd and even numbers, fractions, and basic arithmetic operations. Understanding these concepts is crucial for solving problems related to numbers in competitive exams and real-life situations.

1. What is the smallest prime number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The smallest prime number is 2.

2. Find the sum of the first 10 natural numbers.

Sum of the first n natural numbers = n(n + 1)/2.
For the first 10 numbers, sum = 10 * (10 + 1) / 2 = 55.

3. If a number is divided by 6, it leaves a remainder of 3. What is the remainder when the square of the same number is divided by 6?

Let the number be x. When x is divided by 6, the remainder is 3, so x = 6k + 3 for some integer k.
The square of the number is (6k + 3)² = 36k² + 36k + 9 = 6(6k² + 6k + 1) + 3.
When (6k + 3)² is divided by 6, the remainder is 3.

4. What is the least number that must be added to 1056 to make it a perfect square?

The next perfect square after 1056 is 1089 (which is 33²).
Difference = 1089 - 1056 = 33.

5. How many factors does the number 28 have?

Factors of 28 are 1, 2, 4, 7, 14, and 28. So, it has 6 factors.

6. If the product of three consecutive positive integers is 120, what is the largest integer?

Let the integers be n, n+1, and n+2.
n(n+1)(n+2) = 120.
Solving, we find n = 3, n+1 = 4, n+2 = 5.
The largest integer is 5.

7. What is the sum of all prime numbers between 30 and 40?

Prime numbers between 30 and 40 are 31, 37.
Sum = 31 + 37 = 68.

8. Find the smallest number which when divided by 20, 25, 35, and 40 leaves a remainder of 14 in each case.

The number must be LCM(20, 25, 35, 40) - Remainder.
LCM(20, 25, 35, 40) = 1400.
Number = 1400 - 14 = 1386.

9. What is the average of the first 20 even numbers?

Average = (Sum of all numbers) / (Number of items).
Sum of first n even numbers = n(n+1).
For the first 20 numbers, sum = 20 * 21 = 420.
Average = 420 / 20 = 21.

10. If 324 is divided by a certain number, the quotient is 12. What is the divisor?

Let the divisor be x.
324 / x = 12.
x = 324 / 12 = 27.

11. What is the smallest 3-digit prime number?

The smallest 3-digit number is 100. The next prime number after 100 is 101.

12. The sum of two consecutive even numbers is 46. What are the numbers?

Let the first even number be x. Then the next even number is x + 2.
x + (x + 2) = 46.
2x + 2 = 46.
2x = 44.
x = 22.
So, the numbers are 22 and 24.

13. Find the greatest common divisor of 48 and 64.

The greatest common divisor (GCD) of 48 and 64 is the largest number that divides both. The GCD of 48 and 64 is 16.

14. A number when divided by 342 gives a remainder 47. What remainder will be obtained by dividing the same number by 19?

Let the number be x. x = 342k + 47 for some integer k.
Since 342 is a multiple of 19, x mod 19 = 47 mod 19 = 9.

15. How many prime numbers are there between 1 and 100?

There are 25 prime numbers between 1 and 100.

16. What is the average of the first 100 odd numbers?

Average = (Sum of all numbers) / (Number of items).
Sum of the first n odd numbers = n².
For the first 100 numbers, sum = 100² = 10000.
Average = 10000 / 100 = 100.

17. If 45 is subtracted from two-thirds of a number, the result is 55. What is the number?

Let the number be x.
(2/3)x - 45 = 55.
(2/3)x = 100.
x = 100 * (3/2) = 150.

18. The least number that must be subtracted from 1340 so that the remainder is divisible by 12 is:

1340 mod 12 = 4. So, subtracting 4 from 1340 makes it divisible by 12.

19. What is the largest 4-digit number that is divisible by 88?

The largest 4-digit number is 9999. Dividing 9999 by 88 gives a remainder of 55. So, the largest 4-digit number divisible by 88 is 9999 - 55 = 9944.

20. A number, when divided by 210, gives a remainder of 55. What remainder will be obtained by dividing the same number by 35?

Let the number be x. x = 210k + 55 for some integer k.
Since 210 is a multiple of 35, x mod 35 = 55 mod 35 = 20.