A number system in mathematics is a writing system for expressing numbers. It is a mathematical notation for representing numbers of a given set, using digits or symbols in a consistent manner. The most common number systems include:

**Decimal Number System (Base-10)**:

- Uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
- The position of each digit represents a power of 10.
- Example: 345 in decimal means 3×102+4×101+5×1003×102+4×101+5×100.

**Binary Number System (Base-2)**:

- Uses two digits: 0 and 1.
- Each position represents a power of 2.
- Example: 1011 in binary means 1×23+0×22+1×21+1×201×23+0×22+1×21+1×20, which equals 11 in decimal.

**Octal Number System (Base-8)**:

- Uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7.
- Each position represents a power of 8.
- Example: 157 in octal means 1×82+5×81+7×801×82+5×81+7×80, which equals 111 in decimal.

**Hexadecimal Number System (Base-16)**:

- Uses sixteen symbols: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15).
- Each position represents a power of 16.
- Example: 1A3 in hexadecimal means 1×162+10×161+3×1601×162+10×161+3×160, which equals 419 in decimal.

**Roman Numerals**:

- Uses combinations of letters from the Latin alphabet: I, V, X, L, C, D, M.
- Example: XVII means 10 + 5 + 1 + 1 = 17.

**Natural Numbers**:

- Positive integers starting from 1, 2, 3, and so on.
- Sometimes 0 is included in this set.

**Rational Numbers**:

- Numbers that can be expressed as the quotient or fraction 𝑝/𝑞, where 𝑝 and 𝑞 are integers and 𝑞≠0.

**Irrational Numbers**:

- Numbers that cannot be expressed as a simple fraction.
- Their decimal expansion is non-repeating and non-terminating, e.g., 𝜋, under root 2.

**Real Numbers**:

- All numbers on the number line, including both rational and irrational numbers.

**Complex Numbers**:

- Numbers in the form 𝑎+𝑏𝑖, where 𝑎 and 𝑏 are real numbers and 𝑖 is the imaginary unit, with the property that 𝑖
^{2}= -1.

**Number System Chart**

**Solved Examples of Questions based on Number System**

**Q.1. The sum of two numbers is 156 and their HCF is 13. The numbers of such number pair is?**

**Solution.**

Given:

Sum of the numbers = 156

H.C.F of numbers = 13

Concept used:

If the H.C.F of two numbers is H, then the numbers are Hx and Hy. Where x and y are relatively prime numbers or coprimes.

Calculation:

Let numbers are 13x and 13y, where x and y are coprime

Now, 13x + 13y = 156

13 (x + y) = 156

x+y =156/13 = 12

.. possible values if x and y are

(x = 1, y = 11)

(x = 5, y = 7)

Therefore, possible numbers of pairs is 2

**Q.2. Find the sum of the factors of 3240.**

**Solution.**

Concept:

a, and b must be prime number

Sum of all factors = (a + a + a² + ….. + a) (b + b² + b² + …… + b)

Solution:

3240 23 x 34 x 51 Sum of factors = (2° +21+22+23) (30+31+32+33+34) (50+51) (1+2+4+8) (1+3+9+27+81) (1+5) 15 x 121 x 6 = 10890

.. required sum is 10890

**Q.3. Four bells ring simultaneously at starting and an interval of 6 sec, 12 sec, 15 sec and 20 sec respectively. How many times they ring together in 2 hours?**

**Solution.**

Concept: LCM: It is a number which is a multiple of two or more numbers.

CALCULATION:

LCM of (6, 12, 15, 20) = 60

All 4 bells ring together again after every 60 seconds

Now,

In 2 Hours, they ring together = [(2 x 60 x 60)/60] times + 1 (at the starting) = 121 times

: In 2 hours they ring together for 121 times

**Q.4. If a number is in the form of 810 x 97 x 78, find the total number of prime factors of the given number.**

**Solution**.

Concept used: If a number of the form xª * y^{b * }z^{c} and so on, then total prime factors = a + b + c ……

Where x, y, z, … are prime numbers

Calculation:

The number 810 x 97 x 78 can be written as (23) 10 x (37)7 x 78

The number can ve written as 230 x 314 x 78

Total number of prime factors = 30 + 14 + 8

.: The total number of prime factors are 52

**Q.5. If the 5-digit number 676xy is divisible by 3, 7 and 11, then what is the value of (3x – 5y)?**

**Solution.**

Given:

676xy is divisible by 3, 7 & 11

Concept:

When 676xy is divisible by 3, 7 &11, it will also be divisible by the LCM of 3, 7 &11.

Dividend = Divisor x Quotient + Remainder

Calculation:

LCM (3, 7, 11) = 231

By taking the largest 5-digit number 67699 and divide it by 231.

67699 = 231 x 293 + 16

67699 67683 + 16

67699-1667683 (completely divisible by 231)

. 67683 = 676xy (where x = 8, y = 3)

(3x-5y) = 3 x 8-5×3

24-15=9

.: The required result = 9

**Q.6. How many multiples of both 3 and 4 are there from 1 to 100 in total?**

**Solution.**

Formula used:

n(AUB) = n(A) + n(B) n(A∩B)

Calculation:

On dividing 100 by 3 we get a quotient of 33

The number of multiple of 3, n(A) = 33

On dividing 100 by 4 we get a quotient of 25

The number of multiple of 4, n(B) = 25

LCM of 3 and 4 is 12

On dividing 100 by 12 we get a quotient of 8

The number of multiple of 12, n(AB) = 8

The number which is multiple of 3 or 4 = n(AUB)

Now, n(AUB)=n(A) + n(B) n(A∩B)

33+25-8

⇒50

.. The total number multiple of 3 or 4 is 50

**Q.7. Which of the following is a divisor of (49 ^{15} – 1)?**

**Solution.**

Given:

(49151)

Concept used:

an bn is divisible by (a + b) when n is an even positive integer.

Here, a & b should be prime number.

Calculation:

(4915-1)

=((72)15-1)

=(730-1)

Here, 30 is a positive integer.

According to the concept, (7301) is divisible by (7 + 1) i.e., 8. .. 8 is a divisor of (49151).

**Q.8. Insert 3 rational numbers between 4 & 5.**

**a) 9/25 (b) 37/78**

**Solution: **

In 9/25, the prime factors of denominator are 25 are 5,5,. Thus, it is a terminating decimal.

In 37/78, the prime factors of denominator are 2,3 and 13. Thus, it is a non-terminating decimal.

**Q.9. Find 4 rational numbers between 1 & 2.**

**Solution.**

To find 4 rational numbers between 1 and 2, we need to divide and multiply both the numbers by (4 + 1) which is 5. So we get,

1* 5/5= 5/5 and 2* 5/5= 10/5

Therefore the rational numbers are:

5/5, 6/5, 7/5, 8/5, 9/5, 10/5.

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