Number System Aptitude Formula
- Prime Numbers: Prime numbers are positive integers greater than 1 that have no divisors other than 1 and themselves. Examples include 2, 3, 5, 7, 11, etc.
- Composite Numbers: Composite numbers are positive integers greater than 1 that are not prime. They have divisors other than 1 and themselves.
- Factors: Factors of a number are the positive integers that divide the number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
- Multiples: Multiples of a number are obtained by multiplying that number by any positive integer. For example, the multiples of 5 are 5, 10, 15, 20, etc.
- Divisibility Rules:
- A number is divisible by 2 if the last digit is even (0, 2, 4, 6, or 8).
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- A number is divisible by 4 if the last two digits form a number divisible by 4.
- A number is divisible by 5 if the last digit is 0 or 5.
- A number is divisible by 6 if it is divisible by both 2 and 3.
- A number is divisible by 9 if the sum of its digits is divisible by 9.
- A number is divisible by 10 if the last digit is 0.
- Least Common Multiple (LCM): The LCM of two or more numbers is the smallest number that is a multiple of all those numbers. You can find LCM using prime factorization or the method of taking the highest powers of the prime factors.
- Greatest Common Divisor (GCD): The GCD of two or more numbers is the largest number that divides all of those numbers. You can find GCD using prime factorization or the Euclidean algorithm.
- Fractions:
- A fraction is represented as a/b, where ‘a’ is the numerator, and ‘b’ is the denominator.
- To simplify a fraction, find the GCD of the numerator and denominator and divide both by it.
- Decimals:
- Converting a fraction to a decimal: Divide the numerator by the denominator.
- Converting a decimal to a fraction: Write the decimal as a fraction over a power of 10.
- Binary Numbers:
- Binary is a base-2 number system, consisting of 0s and 1s.
- To convert a decimal number to binary, repeatedly divide by 2 and note the remainders.
- To convert a binary number to decimal, multiply each digit by the corresponding power of 2 and sum the results.
Number System Aptitude Questions with Solution
Question 1: Find the least common multiple (LCM) of 12 and 18.
Solution: To find the LCM of 12 and 18, we can use the prime factorization method.
Prime factors of 12: 2^2 * 3 Prime factors of 18: 2 * 3^2
To find the LCM, we take the highest power of each prime factor:
LCM = 2^2 * 3^2 = 4 * 9 = 36
So, the LCM of 12 and 18 is 36.
Question 2: If a number is divisible by both 4 and 5, what is the smallest positive number it can be divisible by?
Solution: To find the smallest number divisible by both 4 and 5, we need to find their least common multiple (LCM).
Prime factors of 4: 2^2 Prime factors of 5: 5
To find the LCM, we take the highest power of each prime factor:
LCM = 2^2 * 5 = 4 * 5 = 20
So, the smallest number divisible by both 4 and 5 is 20.
Question 3: What is the remainder when 1264 is divided by 9?
Solution: To find the remainder when 1264 is divided by 9, we can use the concept of the remainder theorem. The remainder when a number is divided by 9 is equal to the remainder when the sum of its digits is divided by 9.
Sum of the digits of 1264 = 1 + 2 + 6 + 4 = 13
Now, find the remainder when 13 is divided by 9:
13 ÷ 9 = 1 with a remainder of 4
So, the remainder when 1264 is divided by 9 is 4.
These are just a few examples of number system aptitude questions. You can practice more questions and explore various concepts within the number system to improve your aptitude skills.
I can provide you with a list of 10 number system aptitude questions along with their solutions. If you need more questions, please let me know. Here are the first 10 questions:
Question 1: Find the least common multiple (LCM) of 8 and 12.
Solution: Prime factors of 8: 2^3 Prime factors of 12: 2^2 * 3
To find the LCM, take the highest power of each prime factor:
LCM = 2^3 * 3 = 8 * 3 = 24
So, the LCM of 8 and 12 is 24.
Question 2: What is the smallest positive number divisible by both 7 and 9?
Solution: To find the smallest number divisible by both 7 and 9, calculate their LCM.
Prime factors of 7: 7 Prime factors of 9: 3^2
LCM = 7 * 3^2 = 7 * 9 = 63
So, the smallest number divisible by both 7 and 9 is 63.
Question 3: What is the remainder when 253 is divided by 8?
Solution: To find the remainder when 253 is divided by 8, perform the division:
253 ÷ 8 = 31 with a remainder of 5
So, the remainder when 253 is divided by 8 is 5.
Question 4: Find the greatest common divisor (GCD) of 24 and 36.
Solution: To find the GCD of 24 and 36, use the prime factorization method.
Prime factors of 24: 2^3 * 3 Prime factors of 36: 2^2 * 3^2
To find the GCD, take the lowest power of each prime factor:
GCD = 2^2 * 3 = 4 * 3 = 12
So, the GCD of 24 and 36 is 12.
Question 5: Express the fraction 7/15 in its simplest form.
Solution: To simplify the fraction 7/15, find the GCD of 7 and 15:
GCD(7, 15) = 1
Now, divide both the numerator and denominator by the GCD:
(7 ÷ 1) / (15 ÷ 1) = 7/15
So, the simplest form of 7/15 is 7/15.
Question 6: Find the binary representation of the decimal number 25.
Solution: To convert the decimal number 25 to binary, repeatedly divide by 2 and note the remainders:
25 ÷ 2 = 12 remainder 1 12 ÷ 2 = 6 remainder 0 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1
Now, write down the remainders in reverse order to get the binary representation:
25 in binary is 11001.
Question 7: If a number is divisible by both 8 and 10, what is the smallest positive number it can be divisible by?
Solution: To find the smallest number divisible by both 8 and 10, calculate their LCM:
Prime factors of 8: 2^3 Prime factors of 10: 2 * 5
LCM = 2^3 * 5 = 8 * 5 = 40
So, the smallest number divisible by both 8 and 10 is 40.
Question 8: What is the sum of the first 20 natural numbers?
Solution: The sum of the first n natural numbers is given by the formula (n * (n + 1)) / 2.
For n = 20:
Sum = (20 * (20 + 1)) / 2 = (20 * 21) / 2 = 210
So, the sum of the first 20 natural numbers is 210.
Question 9: Find the remainder when 999 is divided by 7.
Solution: To find the remainder when 999 is divided by 7, perform the division:
999 ÷ 7 = 142 with a remainder of 5
So, the remainder when 999 is divided by 7 is 5.
Question 10: Express the decimal number 0.625 as a fraction in simplest form.
Solution: To express the decimal 0.625 as a fraction, write it as 625/1000 and simplify:
0.625 = 625/1000
Now, find the GCD of 625 and 1000:
GCD(625, 1000) = 125
Divide both the numerator and denominator by the GCD:
(625 ÷ 125) / (1000 ÷ 125) = 5/8
So, 0.625 as a fraction in simplest form is 5/8.
Question 11: Find the sum of the first 15 even natural numbers.
Solution: The first 15 even natural numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, and 30.
To find their sum, you can use the formula for the sum of an arithmetic series:
Sum = (n/2) * [2a + (n-1)d]
Where:
- n is the number of terms (15 in this case).
- a is the first term (2 in this case).
- d is the common difference (2 in this case).
Sum = (15/2) * [2 * 2 + (15 – 1) * 2] Sum = (15/2) * [4 + 28] Sum = (15/2) * 32 Sum = 15 * 16 Sum = 240
So, the sum of the first 15 even natural numbers is 240.
Question 12: Express the fraction 9/12 in its simplest form.
Solution: To simplify the fraction 9/12, find the GCD of 9 and 12:
GCD(9, 12) = 3
Now, divide both the numerator and denominator by the GCD:
(9 ÷ 3) / (12 ÷ 3) = 3/4
So, the simplest form of 9/12 is 3/4.
Question 13: Find the remainder when 5678 is divided by 11.
Solution: To find the remainder when 5678 is divided by 11, perform the division:
5678 ÷ 11 = 516 with a remainder of 2
So, the remainder when 5678 is divided by 11 is 2.
Question 14: What is the largest 3-digit number that is divisible by both 4 and 6?
Solution: To find the largest 3-digit number divisible by both 4 and 6, calculate their LCM:
Prime factors of 4: 2^2 Prime factors of 6: 2 * 3
LCM = 2^2 * 3 = 4 * 3 = 12
Now, find the largest 3-digit multiple of 12:
The largest 3-digit number is 999.
999 divided by 12 is 83 with a remainder of 3.
So, the largest 3-digit number divisible by both 4 and 6 is 999 – 3 = 996.
Question 15: Convert the binary number 101011 to its decimal equivalent.
Solution: To convert the binary number 101011 to decimal, write down the powers of 2 and multiply by the corresponding binary digit:
1 * 2^5 + 0 * 2^4 + 1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0
32 + 0 + 8 + 0 + 2 + 1 = 43
So, the decimal equivalent of the binary number 101011 is 43.
Question 16: Find the GCD of 36 and 48.
Solution: To find the GCD of 36 and 48, use the prime factorization method.
Prime factors of 36: 2^2 * 3^2 Prime factors of 48: 2^4 * 3
To find the GCD, take the lowest power of each prime factor:
GCD = 2^2 * 3 = 4 * 3 = 12
So, the GCD of 36 and 48 is 12.
Question 17: Express the decimal number 0.75 as a fraction in simplest form.
Solution: To express the decimal 0.75 as a fraction, write it as 75/100 and simplify:
0.75 = 75/100
Now, find the GCD of 75 and 100:
GCD(75, 100) = 25
Divide both the numerator and denominator by the GCD:
(75 ÷ 25) / (100 ÷ 25) = 3/4
So, 0.75 as a fraction in simplest form is 3/4.
Question 18: What is the remainder when 789 is divided by 13?
Solution: To find the remainder when 789 is divided by 13, perform the division:
789 ÷ 13 = 60 with a remainder of 9
So, the remainder when 789 is divided by 13 is 9.
Question 19: Find the least common multiple (LCM) of 15 and 20.
Solution: To find the LCM of 15 and 20, use the prime factorization method.
Prime factors of 15: 3 * 5 Prime factors of 20: 2^2 * 5
To find the LCM, take the highest power of each prime factor:
LCM = 2^2 * 3 * 5 = 4 * 3 * 5 = 60
So, the LCM of 15 and 20 is 60.
Question 20: If a number is divisible by both 3 and 5, what is the smallest positive number it can be divisible by?
Solution: To find the smallest number divisible by both 3 and 5, calculate their LCM:
Prime factors of 3: 3 Prime factors of 5: 5
LCM = 3 * 5 = 15
So, the smallest number divisible by both 3 and 5 is 15.
Question 21: What is the remainder when 1357 is divided by 9?
Solution: To find the remainder when 1357 is divided by 9, you can use the concept of the remainder theorem. The remainder when a number is divided by 9 is equal to the remainder when the sum of its digits is divided by 9.
Sum of the digits of 1357 = 1 + 3 + 5 + 7 = 16
Now, find the remainder when 16 is divided by 9:
16 ÷ 9 = 1 with a remainder of 7
So, the remainder when 1357 is divided by 9 is 7.
Question 22: Find the GCD of 72 and 90.
Solution: To find the GCD of 72 and 90, use the prime factorization method.
Prime factors of 72: 2^3 * 3^2 Prime factors of 90: 2 * 3^2 * 5
To find the GCD, take the lowest power of each prime factor:
GCD = 2 * 3^2 = 2 * 9 = 18
So, the GCD of 72 and 90 is 18.
Question 23: Express the fraction 15/20 in its simplest form.
Solution: To simplify the fraction 15/20, find the GCD of 15 and 20:
GCD(15, 20) = 5
Now, divide both the numerator and denominator by the GCD:
(15 ÷ 5) / (20 ÷ 5) = 3/4
So, the simplest form of 15/20 is 3/4.
Question 24: Convert the octal number 345 to its decimal equivalent.
Solution: To convert the octal number 345 to decimal, multiply each digit by the corresponding power of 8 and sum the results:
3 * 8^2 + 4 * 8^1 + 5 * 8^0
192 + 32 + 5 = 229
So, the decimal equivalent of the octal number 345 is 229.
Question 25: Find the sum of the first 10 odd natural numbers.
Solution: The first 10 odd natural numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19.
To find their sum, you can use the formula for the sum of an arithmetic series:
Sum = (n/2) * [2a + (n-1)d]
Where:
- n is the number of terms (10 in this case).
- a is the first term (1 in this case).
- d is the common difference (2 in this case).
Sum = (10/2) * [2 * 1 + (10 – 1) * 2] Sum = 5 * [2 + 18] Sum = 5 * 20 Sum = 100
So, the sum of the first 10 odd natural numbers is 100.
Question 26: What is the largest 3-digit number that is divisible by 9?
Solution: To find the largest 3-digit number divisible by 9, you can divide 999 (the largest 3-digit number) by 9:
999 ÷ 9 = 111
So, the largest 3-digit number divisible by 9 is 111.
Question 27: Express the fraction 12/15 in its simplest form.
Solution: To simplify the fraction 12/15, find the GCD of 12 and 15:
GCD(12, 15) = 3
Now, divide both the numerator and denominator by the GCD:
(12 ÷ 3) / (15 ÷ 3) = 4/5
So, the simplest form of 12/15 is 4/5.
Question 28: Find the remainder when 7654 is divided by 12.
Solution: To find the remainder when 7654 is divided by 12, perform the division:
7654 ÷ 12 = 637 with a remainder of 10
So, the remainder when 7654 is divided by 12 is 10.
Question 29: Find the least common multiple (LCM) of 6 and 9.
Solution: To find the LCM of 6 and 9, use the prime factorization method.
Prime factors of 6: 2 * 3 Prime factors of 9: 3^2
To find the LCM, take the highest power of each prime factor:
LCM = 2 * 3^2 = 2 * 9 = 18
So, the LCM of 6 and 9 is 18.
Question 30: If a number is divisible by both 10 and 15, what is the smallest positive number it can be divisible by?
Solution: To find the smallest number divisible by both 10 and 15, calculate their LCM:
Prime factors of 10: 2 * 5 Prime factors of 15: 3 * 5
LCM = 2 * 3 * 5 = 30
So, the smallest number divisible by both 10 and 15 is 30.
50 Number System Objective Questions with Answer
I can provide you with 10 multiple-choice number system objective questions along with their answers. If you need more questions, please let me know.
Question 1: What is the remainder when 1536 is divided by 8?
a) 0 b) 2 c) 4 d) 6
Answer: b) 2
Question 2: Which of the following is the largest prime number?
a) 7 b) 21 c) 13 d) 1
Answer: c) 13
Question 3: What is the binary representation of the decimal number 25?
a) 10101 b) 11001 c) 11110 d) 10011
Answer: b) 11001
Question 4: What is the GCD (Greatest Common Divisor) of 36 and 48?
a) 4 b) 6 c) 12 d) 18
Answer: c) 12
Question 5: What is the smallest positive number that is divisible by both 3 and 4?
a) 6 b) 9 c) 10 d) 12
Answer: a) 6
Question 6: If a number is divisible by both 7 and 11, what is the smallest positive number it can be divisible by?
a) 7 b) 11 c) 49 d) 77
Answer: d) 77
Question 7: What is the remainder when 978 is divided by 13?
a) 5 b) 8 c) 1 d) 0
Answer: b) 8
Question 8: What is the LCM (Least Common Multiple) of 6 and 8?
a) 12 b) 18 c) 24 d) 48
Answer: a) 12
Question 9: Express the decimal number 0.625 as a fraction in simplest form.
a) 5/8 b) 1/4 c) 25/40 d) 10/16
Answer: a) 5/8
Question 10: What is the sum of the first 20 natural numbers?
a) 210 b) 220 c) 230 d) 240
Answer: d) 240
Question 11: What is the remainder when 2567 is divided by 9?
a) 0 b) 2 c) 4 d) 5
Answer: d) 5
Question 12: Which of the following is a perfect square?
a) 14 b) 25 c) 17 d) 33
Answer: b) 25
Question 13: What is the binary representation of the decimal number 45?
a) 101101 b) 110101 c) 111011 d) 100101
Answer: a) 101101
Question 14: What is the GCD (Greatest Common Divisor) of 56 and 72?
a) 4 b) 8 c) 12 d) 16
Answer: b) 8
Question 15: What is the smallest positive number that is divisible by both 5 and 9?
a) 15 b) 27 c) 45 d) 90
Answer: a) 15
Question 16: If a number is divisible by both 6 and 8, what is the smallest positive number it can be divisible by?
a) 12 b) 24 c) 48 d) 72
Answer: a) 12
Question 17: What is the remainder when 7834 is divided by 7?
a) 2 b) 3 c) 4 d) 5
Answer: b) 3
Question 18: What is the LCM (Least Common Multiple) of 12 and 15?
a) 24 b) 30 c) 60 d) 72
Answer: b) 30
Question 19: Express the decimal number 0.75 as a fraction in simplest form.
a) 1/4 b) 3/5 c) 5/8 d) 2/3
Answer: c) 3/4
Question 20: What is the sum of the first 15 natural numbers?
a) 105 b) 120 c) 136 d) 210
Answer: b) 120
Question 21: What is the remainder when 369 is divided by 12?
a) 3 b) 6 c) 9 d) 12
Answer: b) 6
Question 22: Which of the following is not a prime number?
a) 17 b) 21 c) 23 d) 29
Answer: b) 21
Question 23: What is the binary representation of the decimal number 18?
a) 10010 b) 11000 c) 10101 d) 11110
Answer: a) 10010
Question 24: What is the GCD (Greatest Common Divisor) of 56 and 63?
a) 7 b) 8 c) 9 d) 10
Answer: a) 7
Question 25: What is the smallest positive number that is divisible by both 4 and 5?
a) 4 b) 5 c) 10 d) 20
Answer: c) 10
Question 26: If a number is divisible by both 3 and 7, what is the smallest positive number it can be divisible by?
a) 3 b) 7 c) 14 d) 21
Answer: c) 14
Question 27: What is the remainder when 9721 is divided by 11?
a) 0 b) 1 c) 2 d) 3
Answer: b) 1
Question 28: What is the LCM (Least Common Multiple) of 9 and 12?
a) 12 b) 18 c) 27 d) 36
Answer: d) 36
Question 29: Express the decimal number 0.6 as a fraction in simplest form.
a) 1/5 b) 3/5 c) 2/5 d) 5/3
Answer: c) 2/5
Question 30: What is the sum of the first 25 natural numbers?
a) 300 b) 325 c) 350 d) 375
Answer: b) 325
Question 31: What is the remainder when 4893 is divided by 7?
a) 1 b) 2 c) 3 d) 4
Answer: c) 3
Question 32: Which of the following numbers is a multiple of 9?
a) 17 b) 27 c) 37 d) 47
Answer: b) 27
Question 33: What is the binary representation of the decimal number 63?
a) 110001 b) 100011 c) 111111 d) 101000
Answer: c) 111111
Question 34: What is the GCD (Greatest Common Divisor) of 60 and 72?
a) 6 b) 8 c) 10 d) 12
Answer: d) 12
Question 35: What is the smallest positive number that is divisible by both 6 and 7?
a) 12 b) 14 c) 21 d) 42
Answer: c) 21
Question 36: If a number is divisible by both 4 and 9, what is the smallest positive number it can be divisible by?
a) 18 b) 36 c) 72 d) 144
Answer: a) 18
Question 37: What is the remainder when 7249 is divided by 8?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Question 38: What is the LCM (Least Common Multiple) of 15 and 18?
a) 27 b) 45 c) 54 d) 90
Answer: c) 54
Question 39: Express the decimal number 0.875 as a fraction in simplest form.
a) 1/4 b) 3/8 c) 5/8 d) 7/8
Answer: d) 7/8
Question 40: What is the sum of the first 30 natural numbers?
a) 435 b) 450 c) 465 d) 480
Answer: d) 480
Question 41: What is the remainder when 5694 is divided by 5?
a) 0 b) 1 c) 2 d) 4
Answer: b) 4
Question 42: Which of the following numbers is a multiple of 10?
a) 13 b) 30 c) 21 d) 42
Answer: b) 30
Question 43: What is the binary representation of the decimal number 37?
a) 100101 b) 101001 c) 101010 d) 110001
Answer: a) 100101
Question 44: What is the GCD (Greatest Common Divisor) of 40 and 48?
a) 4 b) 8 c) 12 d) 16
Answer: b) 8
Question 45: What is the smallest positive number that is divisible by both 8 and 12?
a) 8 b) 12 c) 16 d) 24
Answer: d) 24
Question 46: If a number is divisible by both 3 and 6, what is the smallest positive number it can be divisible by?
a) 6 b) 9 c) 12 d) 18
Answer: a) 6
Question 47: What is the remainder when 8391 is divided by 9?
a) 2 b) 3 c) 4 d) 6
Answer: b) 3
Question 48: What is the LCM (Least Common Multiple) of 20 and 24?
a) 30 b) 40 c) 60 d) 120
Answer: c) 60
Question 49: Express the decimal number 0.625 as a fraction in simplest form.
a) 1/4 b) 3/4 c) 1/2 d) 5/8
Answer: d) 5/8
Question 50: What is the sum of the first 50 natural numbers?
a) 1200 b) 1250 c) 1275 d) 1300
Answer: c) 1275
Question 51: What is the remainder when 4836 is divided by 6?
a) 0 b) 2 c) 3 d) 4
Answer: a) 0
Question 52: Which of the following numbers is a multiple of 7?
a) 17 b) 28 c) 33 d) 42
Answer: b) 28
Question 53: What is the binary representation of the decimal number 51?
a) 101101 b) 110011 c) 110101 d) 111011
Answer: a) 110011
Question 54: What is the GCD (Greatest Common Divisor) of 54 and 72?
a) 6 b) 8 c) 12 d) 18
Answer: a) 6
Question 55: What is the smallest positive number that is divisible by both 7 and 8?
a) 14 b) 21 c) 28 d) 56
Answer: c) 28
Question 56: If a number is divisible by both 5 and 10, what is the smallest positive number it can be divisible by?
a) 5 b) 10 c) 15 d) 20
Answer: b) 10
Question 57: What is the remainder when 9562 is divided by 7?
a) 0 b) 1 c) 2 d) 3
Answer: a) 0
Question 58: What is the LCM (Least Common Multiple) of 18 and 24?
a) 30 b) 36 c) 48 d) 72
Answer: b) 36
Question 59: Express the decimal number 0.875 as a fraction in simplest form.
a) 1/4 b) 7/8 c) 5/8 d) 3/4
Answer: b) 7/8
Question 60: What is the sum of the first 75 natural numbers?
a) 2475 b) 2800 c) 3000 d) 3150
Answer: c) 3000