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Chapter 15: Playing with Numbers > Finding of Divisibility by taking Remainders of Place Values

Finding of Divisibility by taking Remainders of Place Values

This method helps check divisibility by breaking a number into its place values and analyzing the remainders when each part is divided by a given divisor. Instead of dividing the whole number directly, we look at how each digit contributes to divisibility.

How It Works

  • Break the number into its place values (units, tens, hundreds, etc.).

  • Find the remainder of each place value when divided by the given number.

  • Sum up the remainders to check if the total remainder is divisible by the number.

Example 1: Checking Divisibility of 527 by 7

Instructions

Step 1: Express 527 in place values: 527 = + +
Step 2: Find the remainder of each term when divided by 7: 500 ÷ 7 → remainder = , 20 ÷ 7 → remainder = and 7 ÷ 7 → remainder =
Step 3: Add the remainders 3 + 6 + 0 =
Step 4: Check if 9 is divisible by 7
Since 9 divisible by 7, 527 divisible by 7.

Example 2: Checking Divisibility of 846 by 3.

Instructions

Step 1: Express 846 in place values: 846 = + +
Step 2: Find the remainder of each term when divided by 3: 800 ÷ 3 → remainder = , 40 ÷ 3 → remainder = , 6 ÷ 3 → remainder =
Step 3: Add the remainders: 2 + 1 + 0 =
Step 4: Check if 3 is divisible by 3.
Since, 3 divisible by 3, 846 divisible by 3.

Why Is This Useful?

  • It helps in large numbers where direct division is tough.

  • It provides an alternative approach to checking divisibility.

  • It is useful in modular arithmetic and number theory.