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Divisibility Rules - 10s

Here are some clever rules of divisibility specifically for numbers based on powers of 10 (like 10, 100, 1000, etc.):

Rules of Divisibility for 10s-Based Numbers

  1. Divisibility by 10:

    • A number is divisible by 10 if its last digit is 0.

  2. Divisibility by 100:

    • A number is divisible by 100 if its last two digits are 00.

  3. Divisibility by 1000:

    • A number is divisible by 1000 if its last three digits are 000.

  4. Divisibility by 10,000:

    • A number is divisible by 10,000 if its last four digits are 0000.

  5. Divisibility by 10^n (where \( n \) is any positive integer):

    • A number is divisible by \( 10^n \) if its last \( n \) digits are all zeros.

Patterns and Tricks for 10s-Based Divisibility

  1. Divisibility by 20:

    • A number is divisible by 20 if its last two digits form a number divisible by 20.

  2. Divisibility by 200:

    • A number is divisible by 200 if its last three digits form a number divisible by 200.

  3. Divisibility by 2000:

    • A number is divisible by 2000 if its last four digits form a number divisible by 2000.

  4. Divisibility by 50:

    • A number is divisible by 50 if its last digit is 0 and the second last digit is either 0 or 5.

  5. Divisibility by 500:

    • A number is divisible by 500 if its last three digits are 000, and the digits immediately before the last three are divisible by 5.

General Tips for 10s-Based Divisibility

  1. Multiples of Powers of 10:

    • For any multiple of \( 10^n \), check that the last \( n \) digits of the number are zeros.

  2. Combination of 10s-Based Divisibility:

    • For numbers like 30, 40, 60, etc., you can use a combination of the basic rules. For example, a number is divisible by 30 if it is divisible by both 10 and 3. Similarly, check the divisibility for multiples of 40 by confirming divisibility by 10 and 4.

Examples

  1. Example for Divisibility by 1000:

    • To check if 123,000 is divisible by 1000, look at the last three digits: 000. Since they are all zeros, 123,000 is divisible by 1000.

  2. Example for Divisibility by 200:

    • To check if 45,800 is divisible by 200, check the last three digits: 800. Since 800 is divisible by 200, 45,800 is divisible by 200.

  3. Example for Divisibility by 500:

    • To check if 123,500 is divisible by 500, check the last three digits: 500. Since 500 is divisible by 500, 123,500 is divisible by 500.

Summary

The rules for divisibility by numbers based on powers of 10 are straightforward and involve looking at the number of trailing zeros in a number. For other numbers that are multiples of 10, you combine the basic divisibility rules of their factors. These rules help simplify checking divisibility and solving problems involving large numbers.