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Thought-Process to Discover Knowledge

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mathsDivisibility in Whole NumbersReducing number of Digits for Divisibility Test

### Divisibility: Simplification in Digits

In this page, Simplification of Divisibility test, by reducing the number of digits of the dividend, is explained. This method is used to derive divisibility tests for 7 and 13.

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What is "Divisibility Test: Simplification by Subtraction"?

• difference between the dividend and a multiple of divisor is checked for divisibility by divisor
• difference between the dividend and a multiple of divisor is checked for divisibility by divisor
• dividend and divisor are divided by common factors and the result is checked for divisibility

The answer is "difference between the dividend and a multiple of divisor is checked for divisibility by divisor"

What is "Divisibility Test: Simplification by Division"?

• dividend can be divided by a co-prime (of divisor) factor and result is checked for divisibility
• dividend and divisor are divided by common factors and the result is checked for divisibility
• both the above
• both the above

The answer is "both the above".

The "simplification by subtraction" and "simplification by division" are further refined.

Take a number 2352 and check for divisibility by 7.
We can simplify the number as

divisibility of 2352

subtracting a multiple of 7
divisibility ->2352-7xx3xx2

divisibility ->2352-42 = 2310

2 and 5 are co-primes of divisor 7
divisibility ->2310/10 = 231
repeating subtraction of a multiple of 7
divisibility ->231-7xx3xx1 = 210

repeating 2 and 5 co-prime simplification
divisibility ->210/10 = 21

21 is divisible by 7, and so 2352 is divisible by 7.

Note: The multiple of 7 is chosen with two factors

•  First 3 is chosen because, 7xx3 = 21 where the units place is 1

•  Second 2 is chosen because the units digit of the number is 2 and on subtraction that makes the units digit 0. That is, 2352-21*2 = 2310. This facilitates the co-prime of 10 step, thereby reducing the number of digits.

Another example of combining "simplification by subtraction" and "simplification by division".

Take a number 2352 and check for divisibility by 13.

Look at the multiples of 13 : 26, 39, etc.
The multiple 39 is interesting as 39*n+n is a multiple of 10.

We can simplify the number 2352 as

divisibility of 2352

adding a multiple of 13
divisibility ->2352+13xx3xx2

divisibility ->2352+78 = 2430

2 and 5 are co-primes of divisor 13
divisibility ->2430/10 = 243
repeating addition of a multiple of 13
divisibility ->231+13xx3xx1 = 270

repeating 2 and 5 co-prime simplification
divisibility ->270/10 = 27

27 is not divisible by 13, and so 2352 is not divisible by 13.

Note: The multiple of 13 is chosen with two factors

•  First 3 is chosen because, 13xx3 = 39 where the units place is 9

•  Second 2 is chosen because the units digit of the number is 2 and on addition that makes the units digit 0. That is, 2352+39*2 = 2430. This facilitates the co-prime of 10 step, thereby reducing the number of digits.

Simplification in Digits : Consider the multiples of the divisor and choose the one that ends 1 or 9. By using Simplification by Subtraction and Simplification by Division, reduce the number of digits of the dividend. The reduced dividend can be checked for divisibility, and the simplification can be used repeatedly.

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