NUMBER SYSTEM

 

Number System: Some Important Tricks and Tips for CUET , CLAT and IPMAT Exam
This Article will help you to sharp your skills to solve questions based on Number System .  This Article will help you to clear CUET Exam .

This Article is also Useful for IPMAT.

And also very helpful for Clearing CLAT .
Mastering Number System for CUET, CLAT, and IPMAT: Concepts, Tips, and Previous Year Questions

The Number System is a fundamental chapter in mathematics, crucial for competitive exams like CUET, CLAT, and IPMAT. This blog will cover essential concepts, divisibility rules, unit digits, LCM and HCF, and other related topics. We'll also discuss previous year questions to help you prepare.

 

Divisibility Rules:

1. Divisibility by 2: Last digit is even (0, 2, 4, 6, 8)

2. Divisibility by 3: Sum of digits is divisible by 3

3. Divisibility by 4: Last two digits form a number divisible by 4

4. Divisibility by 5: Last digit is 0 or 5

5. Divisibility by 6: Divisible by both 2 and 3

6. Divisibility by 7: Double the last digit and subtract from remaining digits; if result is divisible by 7, original number is divisible

7. Divisibility by 8: Last three digits form a number divisible by 8

8. Divisibility by 9: Sum of digits is divisible by 9

9. Divisibility by 10: Last digit is 0

 

Tricks:

 

1. Digital Root Method: For divisibility by 3, 6, or 9, find digital root (sum of digits) and check divisibility.

 

Example: Check if 432 is divisible by 3.

Digital root = 4 + 3 + 2 = 9 (divisible by 3)

 

1. Multiply and Subtract Method: For divisibility by 7, multiply last digit by 2 and subtract from remaining digits.

 

Example: Check if 343 is divisible by 7.

Multiply last digit (3) by 2: 6

Subtract from remaining digits (34): 34 - 6 = 28 (divisible by 7)

 

1. Last Digit Focus: For divisibility by 2, 4, 5, or 8, focus only on last digit(s).

 

Example: Check if 1248 is divisible by 8.

Last three digits (248) are divisible by 8.

 

1. Sum of Digits Method: For divisibility by 3, 6, or 9, sum digits and check divisibility.

 

Example: Check if 630 is divisible by 3.

Sum of digits = 6 + 3 + 0 = 9 (divisible by 3)

 

1. Alternate Digit Sum Method: For divisibility by 11, sum alternate digits and check divisibility.

 

Example: Check if 1324 is divisible by 11.

Sum of alternate digits = 1 + 2 - 3 + 4 = 4 (not divisible by 11)

 

Practice Tips:

 

1. Memorize divisibility rules.

2. Practice applying rules to different numbers.

3. Focus on digital root and multiply-subtract methods.

4. Improve mental calculation skills.

5. Solve previous year questions.

Unit Digit:

Cyclicity Patterns:

 

1. Unit digit of powers of 2: 2, 4, 8, 6 (cyclic)

2. Unit digit of powers of 3: 3, 9, 7, 1 (cyclic)

3. Unit digit of powers of 4: 4, 6 (cyclic)

4. Unit digit of powers of 5: Always 5

5. Unit digit of powers of 6: Always 6

6. Unit digit of powers of 7: 7, 9, 3, 1 (cyclic)

7. Unit digit of powers of 8: 8, 4, 2, 6 (cyclic)

8. Unit digit of powers of 9: 9, 1 (cyclic)

 

Unit Digit of Large Numbers:

 

1. Unit digit of (abc)...: Only consider unit digit of 'c'

2. Unit digit of (ab)^n: Consider unit digit of 'b' and 'n'

 

Multiplication Tricks:

 

1. Unit digit of product = Unit digit of product of unit digits

Example: Unit digit of 43 × 27 = Unit digit of 3 × 7 = 1

2. Multiply unit digits of numbers and find unit digit of result

 

Division Tricks:

 

1. Unit digit of quotient = Unit digit of dividend ÷ divisor

Example: Unit digit of 48 ÷ 6 = Unit digit of 8 ÷ 6 = 2

2. Find unit digit of remainder using divisibility rules

 

Power Tricks:

 

1. Unit digit of x^n = Unit digit of x × x × ... (n times)

2. Use cyclicity patterns for powers of 2, 3, 4, 5, 6, 7, 8, 9

 

Solved Examples:

 

1. Find unit digit of 7^2019.

Solution: Unit digit of 7^2019 = Unit digit of 7^3 (cyclic) = 3

2. Find unit digit of 43^25 × 27^12.

Solution: Unit digit of 43^25 = Unit digit of 3^5 = 3

Unit digit of 27^12 = Unit digit of 7^12 = 1

Unit digit of product = Unit digit of 3 × 1 = 3

 

Practice Tips:

 

1. Memorize cyclicity patterns.

2. Practice finding unit digits of large numbers.

3. Focus on multiplication and division tricks.

4. Improve mental calculation skills.

 

LCM (Least Common Multiple) and HCF (Highest Common Factor)

LCM Tricks:

1. Prime Factorization Method: Break down numbers into prime factors to find LCM.

Example: LCM of 12 and 15

12 = 2^2 * 3

15 = 3 * 5

LCM = 2^2 * 3 * 5 = 60

 

1. Listing Multiples Method: List multiples of each number and find the smallest common multiple.

 

Example: LCM of 4 and 6

Multiples of 4: 4, 8, 12, 16, 20, 24

Multiples of 6: 6, 12, 18, 24

LCM = 12

 

1. Shortcut Formula: LCM(a, b) = (a * b) / HCF(a, b)

 

HCF Tricks:

 

1. Prime Factorization Method: Break down numbers into prime factors to find HCF.

 

Example: HCF of 12 and 15

12 = 2^2 * 3

15 = 3 * 5

HCF = 3

 

1. Division Method: Divide larger number by smaller number and find remainder.

 

Example: HCF of 48 and 18

48 ÷ 18 = 2 with remainder 12

18 ÷ 12 = 1 with remainder 6

12 ÷ 6 = 2 with remainder 0

HCF = 6

 

1. Shortcut Formula: HCF(a, b) = (a * b) / LCM(a, b)

 

Common Tricks:

 

1. Identical Numbers: LCM and HCF of identical numbers are the numbers themselves.

2. Co-Prime Numbers: HCF of co-prime numbers is 1.

3. Multiples: LCM of two numbers is always a multiple of both numbers.

 

Practice Tips:

 

1. Practice finding LCM and HCF of different types of numbers (e.g., prime, composite).

2. Use online resources or worksheets for practice.

3. Focus on shortcut formulas and methods.

4. Improve mental calculation skills.

 

Important Formulas:

 

1. LCM(a, b) * HCF(a, b) = a * b

2. LCM(a, b, c) = LCM(LCM(a, b), c)

3. HCF(a, b, c) = HCF(HCF(a, b), c)

Other Important Concepts

 

1. Prime Numbers: Numbers greater than 1, divisible only by 1 and themselves

2. Composite Numbers: Numbers greater than 1, divisible by numbers other than 1 and themselves

3. Co-Prime Numbers: Numbers with HCF = 1

4. Remainder Theorem: Remainder of division of a polynomial by x - a is f(a)

5. Factorial: Product of all positive integers up to n (denoted by n!)

 

Previous Year Questions

 

CUET:

 

1. If 36x81 is divisible by 3, find x. (2022)

2. Find the remainder when 43^25 is divided by 10. (2021)

 

CLAT:

 

1. What is the HCF of 24 and 30? (2022)

2. If x and y are co-prime, find the remainder when x^2 + y^2 is divided by 4. (2021)

 

IPMAT:

 

1. Find the LCM of 12 and 15. (2022)

2. Determine the unit digit of 7^2019. (2021)

 

Tips and Strategies:

1. Practice divisibility rules and unit digit patterns.

2. Memorize LCM and HCF formulas.

3. Understand prime and composite numbers.

4. Apply remainder theorem for polynomial divisions.

5. Use factorial to simplify calculations.

Conclusion

 

Mastering the Number System chapter requires practice and conceptual clarity. Focus on divisibility rules, unit digits, LCM, HCF, and other related topics. Solve previous year questions to improve your problem-solving skills. With consistent practice, you'll excel in CUET  UG,CUET PG ,  CLAT UG , CLAT PG , and IPMAT.