Tricks for maths

Short cut tricks for Mathematics to make calculations easy

Numbers

Numbers: In mathematics numbers are used to count, measure. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, number may refer to a symbol, a word, or a mathematical abstraction.

In Hindu-Arabic system, we have ten digits (0,1,2,3,4,5,6,7,8,9) and a number is denoted by a group of digits, called NUMERAL.

Divisibility test: Check whether a number is divisible by 2, 3, 4, 5, 6, 7, 8, 9 and 11 (here we are assuming that we will get quotient as an integer)

• Divisibility by 0: There is no such a number which can be divided by 0 (zero). Dividing any number by 0 (zero) is such a mathematical operation where you will never get an answer, the answer is Undefined (∞). Even 0/0 is undefined.
                                                                     
  For some concepts follow this link.

• Divisibility by 1: Each and every number is divisible by 1.

Divisibility by 2: A number is divisible by 2 if the last digit of that number is even (i.e. 0, 2, 4, 6 and 8).

Ex:
1234 is a number whose last digit is 4. So it is divisible by 2.

• Divisibility by 3: To check whether a number is divisible by 3 or not, first of all you have to find the sum of all digits in that number. Then divide the sum by 3.

If the sum is divisible by 3, the number is divisible by 3.

If the sum isn’t divisible by 3 then the number isn’t divisible by 3.

EX:

1234 is a number having 4 digits (1, 2, 3 and 4).

Now the sum of those digits is 10 (i.e. 1+2+3+4).

Now the sum (10) is not divisible by 3*, so 1234 is not divisible by 3.

*Again 10 has two digits, those are 1 and 0. Sum of those digits is 1(i.e. 0+1). Clearly 1 is not divisible by 3. So 10 is not divisible by 3.

Second trick, if a number is a multiplication of 3 consecutive numbers then that number is always divisible by 3. This is useful for when the number takes the form of (n × (n − 1) × (n + 1))

123, 234, 345 ….. 789 are such numbers and you can also consider 231, 324… 879 etc.

Third Trick, Let’s us do it another way. Suppose we have a number 113198465130 and we need to check whether it is divisible by 3 or not.

To check we are to subtract the quantity (Let’s say the quantity be Y) of the digits 2, 5, and 8 in the number (113198465130) from the quantity (Let’s say the quantity be X) of the digits 1, 4, and 7 in the number (113198465130). The result must be divisible by 3.

In this example we have four 1s, one 4 and no 7. So we do have five (X) (5) of the digits 1, 4, and 7 in that number (113198465130).

And we have one 5 and one 8 in that number (113198465130). So we have two (Y) of the digits 2, 5, and 8 in the number (113198465130).

So the subtraction X-Y= 5-2=3, which is divisible by 3.

So the number 113198465130 is divisible by 3.

·        Divisibility by 4: Checking divisibility by 4 is very simple. A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

 Ex:

Let’s have a number 1234.

Now the last two digits are 3 and 4. 3 and 4 form thirty four (34).

[While forming number one thing should be kept in mind to keep the sequence in order as it is in that number.]

It is clear that 34 is not divisible by 4.

So the assumed number (1234) is not divisible by 4.

* You can check this by dividing the number two times by 2.

·        Divisibility by 5: Checking the last digit (unit digit) of a number, we can easily determine the divisibility by 5. What we need to do is to check if the unit digit is 0 or 5. If it is. Then the number is divisible by 5.

Ex:

If the last digit is 0

1.   1110 (The original number)

2.   111 0 (Take the last digit of the number, and check if it is 0 or 5)

3.   111 0 (If it is 0, take the remaining digits, discarding the last)

4.   111 × 2 = 222 (Multiply the result by 2)

5.   1110 ÷ 5 = 222 (The result is the same as the original number divided by 5)

* By following up to step 4 we can simply get the answer.

Step 5 is the conventional way we generally do to get answer.

If the last digit is 5

1.   85 (The original number)

2.   8 5 (Take the last digit of the number, and check if it is 0 or 5)

3.   8 5 (If it is 5, take the remaining digits, discarding the last)

4.   8 × 2 = 16 (Multiply the result by 2)

5.   16 + 1 = 17 (Add 1 to the result)

6.   85 ÷ 5 = 17 (The result is the same as the original number divided by 5)

* By following up to step 5 we can simply get the answer.

Step 6 is the conventional way we generally do to get answer.

·        Divisibility by 6: To perform this test, first of all one need to see if the number is an even number or not.

1.   If the number isn’t an even number (i.e. odd number), the original number never be divisible by 6.

2.   If the number is an even number, the number has a tendency to be divisible by 6.

If 2nd step satisfies then follow this last step:

o   Check if the original number is divisible by 3 or not.

o   If the number is divisible by 3, the original number is divisible by 6.

o   If the original number is not divisible by 3, then the original number never going to be divisible by 6.