This page analyses developing divisibility test procedure for numbers given as sum of multiple numbers. This forms the foundation to developing other divisibility tests for specific numbers.

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is `9+6` a number?

- It does not represent a number, unless the addition is simplified.
- It is a number given as an expression
- It is a number given as an expression

The answer is "it is a number given as an expression"

It is known that addends `9` and `6` are divisible by `3`. Is the sum `9+6` divisible by `3`?

- Yes. Since the addends are divisible, the sum is divisible
- Yes. Since the addends are divisible, the sum is divisible
- Without finding the sum `9+6=15`, it is not possible to decide based only on the addends

The answer is "Yes. Since the addends are divisible, the sum is divisible"

It is known that minuend `9` and subtrahend `6` are divisible by `3`. Is the difference `9-6` divisible by `3`?

- Yes. The difference is divisible
- Yes. The difference is divisible
- Without finding the difference, it is not possible to decide

The answer is "Yes. The difference is divisible".

If a number is given as sum of two addends, and one of the addends is divisible, then the divisibility of the sum is decided by the other addend.

eg: Is `6+8` divisible by `3`? Since `6` is divisible by `3`, the divisibility of `6+8` is decided by divisibility of `8` by `3`.

If a number is given as difference of two numbers, and one of the numbers is divisible, then the divisibility of the difference is decided by the other number. eg: Is `15-6` divisible by `3`? Since `6` is divisible by `3`, the divisibility of `15-6` is decided by divisibility of `15` by `3`. eg: Is `3000-23` divisible by `3`? Since `3000` is divisible by `3`, the divisibility of `3000-23` is decided by divisibility of `23` by `3`.

We studied that divisibility test of sum or difference can be simplified into divisibility test on addend or minuend or subtrahend. This property is further developed to simplify divisibility test as follows.

To simplify divisibility test of a number, a multiple of divisor can be added or subtracted. The divisibility test can be performed on the result.

eg: To find if `892` is divisible by `83`, subtract `830` from `892`, and the result `62` is not divisible by `83` and so `892` is not divisible by `83`.

**Property of Divisibility of sum or difference**: If a number is given as sum of two numbers addend1 + addend2. Then if addend1 is divisible, then divisibility of the sum is decided by the divisibility of the addend2.

If a number is given as difference of two numbers (minuend - subtrahend) and one of the numbers is divisible, then the divisibility of the difference is decided by the divisibility of the other number.**Simplification of Divisibility by Addition or Subtraction** : Divisibility test of a divisor on a number can be simplified by adding or subtracting a multiple of the divisor from the number.

Is `34xx21 + 3` divisible by `34`?

- Yes
- No
- No

The answer is "No". The number is given as sum of two numbers. One addend is divisible by `34`. The divisibility is decided by the second addend `3`.

Is `22xx50-22xx23` divisible by `22`?

- Yes
- Yes
- No

The answer is "Yes". Both minuend and subtrahend are multiples of `22`.

*slide-show version coming soon*