**Number System**

**Problems in number usually involve about farming complex statement about a single number and we suppose to find them. This looks tuft but there is a fact that they are easy to solve and get the mark.**

There is a basic need to solve this problem, is called start with “X”

Lets solve this problems :

**Problem 1**: Find the number such that when 15 is subtracted from 7 times the number, the result is 10 more than twice the number.

**Solution**: They given us the statement about “X”

When 15 is subtracted from 7 times the number, the result is 10 more than twice the number

7X – 15 = 2x + 10

Now the 1st step that we do is after framing this you read your problem statement and find out whether this matches it or not.

The problem statement said 15 when subtracted 7 times from number, the result is twice a number and plus 10.

Now solve the equations : 7X – 15 = 2X + 10

7X – 2X = 15 + 10

5X = 25

X = 5

**Problem 2**: The sum of two numbers is 184. If one-third of the one exceeds one seventh of the other by 8, find the smaller number.

**Solution**: _______ + ________ = 184, two no when added, it gives 184.

In the blank space we can use X and Y , but we can eliminate Y.

Consider X + (184 - X), we can simplify it, X & -X get cancel. 184 remain., so why we need Y, when we have X & (184 - X).

X + (184 - X) = 184

Next statement is one-3rd of one Exceeds one seventh of the other by 8

So equation will be... = X/3− ( 184−X )/7=8 (X Divideby 3, minus (184 - X) divide by 7 ),

now solve this two equations.

Take LCM and solve.. 7X−3(184−2)

**/**21 = 8

7X – 552 + 3X = 168.

7X+3X = 168 + 552

10X = 168 + 552

10X = 720

X= 72

And Y = 112

For checking you can add X & Y 72 + 112 = 184

**Some Formulas**

**1. (a+b)^2 = a^2+ 2ab+ b^2**

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

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

here b^2 and -b^2 get cancel and a^2 & -a^2 get cancel.

So the Eq. Will be (a+b)^2 - (a-b)^2 = 4ab.

Now lets solve the 3rd Problem.

**Problem 3 :**If the sum of two number is 42 and their product is 437, then find the absolute difference between the numbers.

**Solution**: we already know that (a+b)^2 - (a-b)^2 = 4ab.

Given that Sum = 42, & Product = 437, they are asking to find Difference between this two no.

We already know this, put the sum = 42 & product = 437 in equation.

(42)^2−(a−b)= 4(437)

1764 – (a−b)2= 1748

1764 – 1748 = (a−b)^2

16 = (a−b)^2 now take the root.

(a - b) = 4

So the Difference is 4.

**Some Concept of Numbers.**

1. 0, 1, 2, 3, ----9. are called digits.

2. 10, 11, 12,----- are called Number.

3. Natural number (N) :- Counting numbers are called natural numbers.

Example:- 1, 2, 3,---etc. are all natural numbers. minimum natural number 1 and maximum natural number ∞

4. Whole numbers (W) :- All counting numbers together with zero from the set of whole numbers Example:- 0, 1, 2, 3, 4, ------ are whole number.

5. Integers (Z) :- All counting numbers, 0 and -ve of counting numbers are called integers.

Example:- -∞---------, -3, -2, -1, 0, 1, 2, 3, -------∞

6. Rational Numbers (Q) :- A Rational Number is a real number that can be written as a simple fraction Example:- {p/q/p,q∈Z}

7. Irrational Numbers :- An Irrational Number is a real number that cannot be written as a simple fraction. Example:- √

8. Even numbers :- A number divisible by 2 is called an even number.

Example:- 0, 2, 4, 6, - - - - - - - - -

9. Odd numbers :- A number not divisible by 2 is called an odd number.

Example:- 1, 3, 5, 7, - - - - - -

10. Composite Numbers :- Numbers greater than 1 which are not prime, are called composite numbers. Example:- 4, 6, 8, 9, 10, - - - -. 6 -> 1,2,3,6.

11. Prime Numbers:- A number greater than 1 having exactly two factors, namely 1 and itself is called a prime number. upto 100 prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97

12. Co-prime Numbers :- Two natural numbers a and b are said to be co-prime if their HCF is 1. Example:- (21, 44), (4, 9), (2, 3), - - - - -

13. Twin prime numbers :- A pair of prime numbers (as 3 and 5 or 11 and 13) differing by two are called twin prime number.

Example:- The twin pair primes between 1 and 100 are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).

14. Face value :- Face value is the actual value of the digit.

Example:- In the number 7635, the "7" has a face value of 7, the face value of 3 is 3 and so on

15. Place value :- The value of where the digit is in the number, such as units, tens, hundreds, etc. Example:- In 352, the place value of the 5 is "tens"

Place value of 2 * 1 = 2;

Place value of 5 * 10 = 50;

Place value of 3 * 100 = 300.

Division Algorithm :- If we divide a number by another number,

then Dividend = (Divisor * Quotient) + Remainder

D = d * Q + R

Example :- 7 = 3 * 2 + 1

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