HEXAGON: An Aid in Remembering the Eight Fundamental Trigonometric Identities

sum up all digits of a number

There are ten digits from 1 to 9 and 0, with help of those digits we  can generate as many  numbers as we like.When we add the digits of any number we get another number, again adding digits of that number we get another number and at last a final digit of between 1 to 9.
Ex 2309876=2+3+0+9+8+7+6=35 , 3+5=8
Now ,  we do some operation like adding or subtractions  or multiplication in these numbers
We get a result and apply the above rule of adding digits for all , we note that in result also satisfies that.
------234 +438=672

• 2+3+4=9
• 4+3+8=15=1+5=6

now add 9+6=15=6

• The result is 672 =6+7+2=15=1+5=6

It means, our result is correct.
For testing accuracy of calculation we can apply this method.

multiplication made simple

multiplication by 5-
If any number is to be multiplied  by 5 then first divide that number by 2 if remainder is 0 put 0 after quotient and if remainder is 1 put 5 after quotient that is the multiplication result.
26 x 5=130
How! divide 26 by 2 quotient is 13 remainder 0 put 0 after 13 that is 130
45x5= 225 - divide 45 by 2 quotient is 22 remainder 1 then put 5 after 22 that is 225.
multiplication by 25-
If any number is divided by 4 there are 4 chances of remainders 0,1,2,and 3  in multiplication of any number by 25 first divide the number by 4 if remainder is 0 put 00 after quotient similarly 25,50 75 if remainders are 1or 2 or 3 respectively Then you can get result easily. example
36 x 25= 900  How! divide 36 by 4 quotient is 9 put 00 as remainder is 0
58 x 25= 1450 divide 58 by 4 quotient is 14 and remainder is 2 put 50 for remainder 2 that is 1350
multiplication by 125-
If any number is to be multiplied  by 125 then first divide that number by 8 if remainder is 0 put 000 after quotient and if remainder is 1 put 125 after quotient if remainder is 2 put 250 and similarly 375,500,625,750,875 for remainder 3,4,5,6 and 7 In this way you can get result quickly.
45 x 125= 5625, 765x 125=95625, 2349 x 125=293625
Try  these- (i)456 x125 =---- (ii)322 x 125=------ (iii)5665 x 125=-------

 In the last post we have done the following-
"Easiest way to multiply 34 by 36 is- first multiply 6 by 4 it is 24 . Then increase 3 by 1 it is 4 . Now multiply 3 by 4 it is 12. Combining  both it is 1224.The required result.

similarly any two digit numbers having tens digits equal and sum of unit digits =10 their multiplication will be as above.

First multiply unit digits. and write the result  then increase tens digit by 1 and multiply both and write combining both we get required result."

 Now suppose the numbers are in reverse order i.e 43 x 63  or unit digits are equal and sum of tens digit is 10.
Then we have to multiply the unit digit  and add unit digit in multiplication of tens.
Thus we get required result 3 x3=09 and 4 x6= 24+3=27 combining both 2709 is the required result.
Do the following  and confirm the result-
67 x 47, 82x22, 74x34 , 91 x11

how to multiply to two digit numbers if

Easiest way to multiply 34 by 36 is- first multiply 6 by 4 it is 24 . Then increase 3 by 1 it is 4 . Now multiply 3 by 4 it is 12. Combining  both it is 1224.The required result.
similarly any two digit numbers having tens digits equal and sum of unit digits =10 their multiplication will be as above.
First multiply unit digits. and write the result  then increase tens digit by 1 and multiply both and write combining both we get required result.
for the following you can do practice.

22x28, 37x33; 67x 63, 89x81 ,45x45,19x11, 69x61
65x65  These tricks are very useful.

Some brain teasing questions

1- A person gives a loan for Rs 1000 on repayment of Rs 100 per month for one year. What is the rate of interest pa ?
2-What is the area of a triangle having sides 5 cm,12,cm and 13 cm.
3-What is the sum of all even numbers from 1 to 100.
4-Which number is neither prime nor composite?
5-What is the sum of all exterior angles of an octagon?

Algebra important formulas

Remember always

(a+b)^2= a^2+b^2+2ab

(a-b)^2= a^2+b^2-2ab

(a+b)(a-b)=a^2-b^2

(a+b)^3= a^3+b^3+3ab(a+b)

 (a-b)^3= a^3-b^3-3ab(a-b)

 a^3+b^3=(a+b)(a^2+b^2-ab)

a^3-b^3=(a-b)(a^2+b^2+ab)

 (a+b)^2= (a-b)^2+4ab

(a-b)^2= (a+b)^2-4ab

 (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ac

(a-b+c)^2=a^2+b^2+c^2-2ab-2bc+2ac

(a-b-c)^2=a^2+b^2+c^2-2ab+2bc-2ac

(x+a)(x+b)=x^2 +x(a+b)+ab

Product of two numbers=  Their LCM   X  Their  HCF