## Study Notes on Important Facts and Formulae

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt. Generally, there are questions asked related to basic concepts, Facts and Formulae of the Arithmetic.

To let you make the most of Mathematics section, we are providing important facts related to the Number System, HCF, LCM and Decimals. At least 5-6 questions are asked from these topics in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

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**FACTORS AND MULTIPLES**: If a number ‘a’ divides another number ‘b’ exactly, then ‘a’ is a factor of ‘b’. In this case, ‘b’ is called a multiple of ‘a’.

** Ex**. Find the total number of factors of 240.

**Sol**. 240 = 2×2×2×2×3×5

** **= 2^{4 }× 3^{1 }× 5^{1}

^{ }^{total factors = 4+1 × 1+1 × 1+1=20}

_{ }

**HCF (Highest common Factor):**The HCF of two or more than two numbers is the greatest number that divides each of them exactly.

**Ex.** HCF of 36, 72, 108

**Sol.** 36 = 2^{2} × 3^{3}^{ }

72 = 2^{3} × 3^{2}

108 = 2^{2 }× 3^{3}

HCF = 2^{2} × 3^{2 }= 36

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**LCM (Lowest Common Multiple):**The least number which is exactly divisible by each one of the given numbers is called their LCM

**Ex.** LCM of 87 and 145

**Sol**. 87= 3×29

145 = 5×29

LCM= 3×5×29= 435

**Product of two numbers =**Products of their HCF and LCM and LCM is always divisible by HCF.

**Ex.** The sum and difference of the LCM and HCF of two numbers are 112 and 72 respectively. If the one of the numbers is 46, find the 2^{nd} number.

** Sol. **

L+H = 112

L-H = 72

___________

2H = 40

___________

H= 20

L+20 = 112

L= 92

1^{st} number = 46

Let the 2^{nd} No. = x

46×x = 92×20

x= 40

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**HCF and LCM of fractions:**

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**HCF of two numbers**= HCF of sum of the numbers and their LCM.

**Decimal fraction:**Fraction in which denominators are power of 10 are known as decimal fractions

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**Recurring Decimal:**If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.

**Basic formulae:**- (a+b) (a-b) = (a²-b²)
- (a-b)² = a²+b²-2ab
- (a+b)²= a²+b²+2ab

**Ex. **(x+y)=21 and xy=108, then what is the value of x²+y²

** Sol.** Sol. x+y= 21

xy= 108

(x+y)² = x²+y²+2xy

(21)² = x²+y²+2×108

441 = x²+y²+216

x²+y² = 225

** Mathematics Study Notes For All Teaching Exams**

- (a+b+c)² = a²+b²+c²+2(ab+bc+ca)
- (a³+b³)= (a+b) (a²+b²-ab)

- (a³-b³)= (a-b) (a²+b²+ab)

- (a³+b³c³-3abc)= (a+b+c) (a²+b²+c²-ab-bc-ca)
- When a+b+c = 0, then a³+b³+c³= 3abc
- (a+b)²+(a-b)²= 2(a²+b²)
- (a-b)³ = a³-b³-3ab(a-b)
- (a+b)²-(a-b)² = 4ab

**BODMAS Rule:**The rule depicts the correct sequence in which operations are to be executed, so as to find out the value of given expression.

**Ex.** ‘B’ for Bracket, ‘O’ for of, ‘D’ for Division, ‘M’ for Multiplication, ‘A’ or Addition and ‘S’ for Subtraction.

** Ex.** The value of (6.5×7.25+8.5×19.5+4.5-8.5)

** Sol. **(6.5×7.25+8.5×19.5+4.5-8.5)

= 47.125+165.75+4.5-8.5

= 217.375-8.5

= 208.875

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