Understanding Divisibility Rules: A Comprehensive Guide

Understanding Divisibility Rules: A Comprehensive Guide

Understanding Divisibility Rules: A Comprehensive Guide

When a number N is divided by a divisor d and leaves no remainder, we say that d completely divides N. Divisibility rules provide a quick way to check whether a number is divisible by another without performing the full division.

In this guide, we’ll cover divisibility rules for common numbers, solved examples, and practical applications. Visual aids such as flowcharts, examples, and tables can be used to enhance understanding.


Divisibility Rules for Common Numbers

Divisibility by 2

  • Rule: Any number ending in 0, 2, 4, 6, or 8 is divisible by 2.
  • Example: 4568 ends with 8, so it is divisible by 2.

Divisibility by 3

  • Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Example: For 65178, the sum of the digits is 6 + 5 + 1 + 7 + 8 = 27, which is divisible by 3.

Divisibility by 4

  • Rule: A number is divisible by 4 if its last two digits form a number divisible by 4.
  • Example: 83764 ends with 64, which is divisible by 4.

Divisibility by 5

  • Rule: Numbers ending in 0 or 5 are divisible by 5.
  • Example: 45, 1890, and 8475 are all divisible by 5.

Divisibility by 6

  • Rule: A number is divisible by 6 if it is divisible by both 2 and 3.
  • Example: 18 is even (divisible by 2) and its digits sum to 9 (divisible by 3). Hence, it is divisible by 6.

Advanced Divisibility Rules

Divisibility by 7

  • Method 1: Subtract twice the last digit from the remaining number. If the result is divisible by 7, the number is also divisible by 7.
    • Example: For 3563119: 356311 – (2 × 9) = 356293. Repeat this process to find divisibility.
  • Method 2: For large numbers, divide them into blocks of three digits from the right and alternate their sums. If the difference between the sums is divisible by 7, the original number is divisible by 7.
    • Example: For 6517739025: Split into blocks 6 | 517 | 739 | 025. Calculate N1 = 6 + 739, N2 = 517 + 25. Check N1 – N2 for divisibility.

Divisibility by 8

  • Rule: A number is divisible by 8 if its last three digits form a number divisible by 8.
  • Example: 83864 ends with 864, which is divisible by 8.

Divisibility by 9

  • Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
  • Example: 65178 has a digit sum of 27, which is divisible by 9.

Divisibility by 10

  • Rule: Numbers ending in 0 are divisible by 10.

Divisibility by 11

  • Rule: Alternately add and subtract the digits of the number. If the result is divisible by 11, the number is divisible by 11.
  • Example: For 1738456038: (1 – 7 + 3 – 8 + 4 – 5 + 6 – 0 + 3 – 8) = -7. Not divisible.

Divisibility by 12

  • Rule: A number divisible by both 3 and 4 is divisible by 12.

Divisibility by 13

  • Method: Similar to Method 2 for divisibility by 7. Divide the number into blocks of three digits and alternate their sums.

Divisibility by 14

  • Rule: Even numbers divisible by 7 are divisible by 14.

Divisibility by 15

  • Rule: Numbers divisible by both 3 and 5 are divisible by 15.

Divisibility by 16

  • Rule: A number is divisible by 16 if its last four digits are divisible by 16.

Solved Examples

  1. Find the largest three-digit number divisible by 7.
    • Solution: Largest three-digit number is 999. Subtract the remainder when 999 is divided by 7. Remainder = 5, so the result is 999 – 5 = 994.
  2. Find the largest six-digit number divisible by 12.
    • Solution: Largest six-digit number is 999999. Subtract the remainder when 999999 is divided by 12. Remainder = 3, so the result is 999999 – 3 = 999996.
  3. Use divisibility rules to check if 833 is divisible by 2, 3, 7, and 9.
    • Divisible by 2: No, 833 is odd.
    • Divisible by 3: No, sum of digits = 14, not divisible by 3.
    • Divisible by 7: Yes, using Method 1, 833 – 6 = 827 – 14 = 813 – 6 = 777, which is divisible by 7.
    • Divisible by 9: No, sum of digits = 14, not divisible by 9.

Conclusion

By following these rules and practicing the examples provided, you can solve divisibility problems quickly and accurately. Let us know if you need further clarification or additional examples!

Understanding Divisibility Rules

 

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