# Australian NSW High School Mathematics

This article discuss Australian NSW High School Mathematics. It covers all areas of the NSW Syllbus topics with tutorials and worked examples in all topics. It will be an essential educational resource for all students.

**Basic Arithmetic and Algebra**

*Basic Arithmetic*

__Divisibility tests__

A number is divisible if its last number is 0, 2, 4, 6, 8. For example if the last digit of 2543784 it is divisible by 2 because the last digit is 4.

A number is divisible by 3 if the addition of its numbers or digits can be divided by 3. For example 3117264 if one wants to determine whether it can be divided by 3 is to add all the digits. In this case add 3+1+1+7+2+6+4 = 24. One can see 24 can be divided by 3 as 3 is a factor of 24. That is 24= 3*8 according to multiplication table.

A number can be divided by 4 if the last two digits can be divided by 4 irrespective of other digits. For example 42254 36 is divisible by 4 as the last two digits 36 is divisible by 4. In this case the last two digits is 36 and it is divisible by 4.

A number can be divisible by 5 if the last digit is 0 or 5. For example 2553460 is divisible by 5 irrespective of other digits because the last digit is 0.

A number is divisible by 6 if it can be divisible by 2 and 3. That is the the number must be even and the addition of these numbers must be divided by 3. For example 2352 is divisible by 6 as it is an even number and the addition of its digits is 12 and it is divisible by 3.

The divisibility of 7 is complex because 7 is a prime number. In Number theory prime number divisibility is separate subject. To test whether a number is divisible by 7 one has to double the last digit and subtract it from the rest of the number and the answer is 0 or divisible by 7 then the number is divisible by 7. It is a continuous process. For example to test whether 23748 is divisible by 7. First step is to double the last digit. In this case 8 must be doubled. That is 16 must be subtract from 2374. In this case 2374- 16 = 2358. 2358 cannot be divided by 7. There fore 23748 cannot be divided by 7.

If a number is even and the last two digits can be divided by 4, then the number can be divided by 8. For example 2334528 can be divided by 8 because the last two digits, that is 28 can be divided by 4 and it is an even number.

If the addition of the digits of a number can be divided by 9 then then the number can be divided by 9. For example 237417 the addition of digits gives 24. In this case 24 is not divisible by 9. There fore applying the rule the number 237417 is not divisible by 9.

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