Number Theory

1. Unit digit
   

   Example 1: Find the unit digit of 252^525.
   Solution: 25=322^5 = 3225=32, so the unit digit is 2.

   Example 2: What is the unit digit of 939^393?
   Solution: 93=7299^3 = 72993=729, so the unit digit is 9.

   Example 3: What is the unit digit of 747^474?
   Solution: 74=24017^4 = 240174=2401, so the unit digit is 1.

2. HCF & LCM-1
   

   Example 1: Find the HCF of 24 and 36.
   Solution: The HCF of 24 and 36 is 12.

   Example 2: Find the LCM of 8 and 12.
   Solution: The LCM of 8 and 12 is 24.

   Example 3: Find the HCF of 45 and 75.
   Solution: The HCF of 45 and 75 is 15.

3. HCF & LCM-2
   

   Example 1: Find the HCF of 16, 24, and 32.
   Solution: The HCF of 16, 24, and 32 is 8.

   Example 2: Find the LCM of 5, 10, and 20.
   Solution: The LCM of 5, 10, and 20 is 20.

   Example 3: What is the LCM of 7 and 14?
   Solution: The LCM of 7 and 14 is 14.

4. Divisibility Rules
   

   Example 1: Is 156 divisible by 3?
   Solution: Yes, because the sum of digits (1 + 5 + 6 = 12) is divisible by 3.

   Example 2: Is 264 divisible by 4?
   Solution: Yes, because the last two digits (64) are divisible by 4.

   Example 3: Is 123 divisible by 5?
   Solution: No, because it does not end in 0 or 5.

5. Divisibility & Divisors
   

   Example 1: How many numbers between 1 and 100 are divisible by 3?
   Solution: The numbers are 3, 6, 9, ..., 99. There are 33 such numbers.

   Example 2: How many numbers between 1 and 50 are divisible by 5?
   Solution: The numbers are 5, 10, 15, ..., 50. There are 10 such numbers.

   Example 3: How many numbers between 1 and 100 are divisible by both 2 and 3?
   Solution: The numbers are divisible by 6. They are 6, 12, 18, ..., 96. There are 16 such numbers.

6. Decimal
7. Exponent
8. Surds
9. Playing with numbers-1
   

   Example 1: Find the sum of the first 5 prime numbers.
   Solution: The first 5 prime numbers are 2, 3, 5, 7, and 11. Their sum is 2+3+5+7+11=28.

   Example 2: What is the product of the first 3 odd numbers?
   Solution: The first 3 odd numbers are 1, 3, and 5. Their product is 1×3×5=15.

   Example 3: What is the sum of the even numbers between 1 and 10?
   Solution: The even numbers between 1 and 10 are 2, 4, 6, 8, and 10. Their sum is 2+4+6+8+10=30.

10. Playing with numbers-2
    

    Example 1: Find the product of all odd numbers between 1 and 10.
    Solution: The odd numbers between 1 and 10 are 1, 3, 5, 7, and 9. Their product is 1×3×5×7×9=945.

    Example 2: What is the sum of the multiples of 5 between 1 and 50?
    Solution: The multiples of 5 between 1 and 50 are 5, 10, 15, ..., 50. Their sum is 5+10+15+20+25+30+35+40+45+50=275.

    Example 3: What is the difference between the largest prime number less than 20 and the smallest prime number greater than 10?
    Solution: The largest prime number less than 20 is 19, and the smallest prime number greater than 10 is 11. The difference is 19−11=8

11. Diophantine Equation-1
    

    Example 1: Solve the Diophantine equation 3x+5y=163x + 5y = 163x+5y=16 for integers x and y.
    Solution: One solution is x=1, y=3.

    Example 2: Find integer solutions to 4x+7y=154x + 7y = 154x+7y=15.
    Solution: One solution is x=1, y=1

    Example 3: Solve 2x+3y=72x + 3y = 72x+3y=7 for integer solutions.
    Solution: One solution is x=2, y=1.

12. Diophantine Equation-2
    

    Example 1: Solve the equation 5x+12y=15x + 12y = 15x+12y=1 for integers x and y
    Solution: One solution is x=−5, y=2.

    Example 2: Find the integer solutions for 7x+9y=357x + 9y = 357x+9y=35.
    Solution: One solution is x=5, y=0.

    Example 3: Solve 6x+11y=16x + 11y = 16x+11y=1 for integers xxx and yyy.
    Solution: One solution is x=2, y=−1.