99-GRE Math-4-Important Key points and formulas

Click here for the previous basics.

In continuation of part - 3, we will see all the important formulas and useful statements which are to be used in the Math test GRE.

Factorization (Continued):

b) Factorization of Polynomial (Continued): 

2) (x + a) (x - b) = x (x - b) + a (x - b)
                          = x ² - b x + a x - ab
                          = x ² + a x - b x - ab
                          = x ² + (a - b) x - ab
Generally, we call x ² as the first term, (a + b) x as the middle term, and ab as the last term. 

Basic concept: 

a) Step-1: See the sign of the last term.
b) Step-2: Here it is "-" so factorize the product of the coefficient of the first term (here it is 1) and the last term in such a way that the DIFFERENCE of these two factors must be the coefficient of the middle term.
c) Step-3: Get the factors.

Example-1:

Factorize: x ² + 4 x - 21.

a) Step-1: Here sign of the last term 21 is "-"
b) Step-2: The coefficient of the first term is 1 and the last term is 21, so the product of 1 and 21 is 21. Now the factors of 21 are 3 and 7 and as the sign of the last term is "-", their subtraction is 7 - 3 = 4 which is the coefficient of the middle term. (Note: Here, the coefficient of the middle term is positive so we took it as 7 - 3).  
                         = x ² + 4 x - 21
                         = x ² + (7 - 3) x - (3 x 7)
                         = x ² + 7 x - 3 x - (3 x 7)
                         = x (x + 7) - 3 (x + 7)
                         = (x - 3) (x + 7)
c) Step-3: So the factors of x ² + 4 x - 21 are (x - 3) and (x + 7) 

Example-2:

Factorize: 8 x ² + 18 x - 5.

a) Step-1: Here sign of the last term 5 is "-"
b) Step-2: The coefficient of the first term is 8 and the last term is 5, so the product of 8 and 5 is 8 X 5. Now the factors of 8 X 5 are 2, 2, 2, and 5 and as the sign of the last term is "-", so, we take the two factors in such a way that their difference will be 18. Here 2, 2, 2, and 5 will give us 2 and 20. So, here subtraction is 20 - 2 = 18 which is the coefficient of the middle term.  
                         = 8 x ² + 18 x - 5
                         = 8 x ² + (20 - 2) x - 5
                         = 8 x ² + 20 x - 2 x - 5
                         = 4 x (2 x + 5) - 1 (2 x + 5)
                         = (2 x + 5) (4 x - 1) 
c) Step-3: So the factors of 8 x ² + 18 x + 5 are (2 x + 5) and (4 x - 1).

In the next part, we will see the remaining 2 types in detail. These 2 types are given below.

3) (x - a) (x + b) = x ² - (a - b) x - ab
4) (x - a) (x - b) = x ² - (a + b) x + ab

In the next part, we will see a few examples and some essential formulae.

Click here for the next basics.

 ANIL SATPUTE

98-GRE Math-3- Important Key points and formulas

Click here for the previous basics.

In continuation of part - 2, we will see all the important formulas and useful statements which are to be used in the Math test GRE.

Factorization (Continued):

b) Factorization of Polynomial: 

Here following 4 possibilities can be studied to understand factorization in a better way.

1) (x + a) (x + b) = x (x + b) + a (x + b)
                          = x ² + b x + a x + ab
                          = x ² + a x + b x + ab
                          = x ² + (a + b) x + ab
2) (x + a) (x - b) = x (x - b) + a (x - b)
                          = x ² - b x + a x - ab
                          = x ² + a x - b x - ab
                          = x ² + (a - b) x - ab
3) (x - a) (x + b) = x (x + b) - a (x + b)
                          = x ² + b x - a x - ab
                          = x ² - a x + b x - ab
                          = x ² - (a - b) x - ab

Note: Here formulae 2 and 3 are of the same type. the coefficient of the middle term is the difference of the Constance and the sign is to be taken from the greater Constance. 
4) (x - a) (x - b) = x (x - b) - a (x - b)
                          = x ² - b x - a x + ab
                          = x ² - a x - b x + ab
                          = x ² - (a + b) x + ab

Now we will study these types in detail:
1) (x + a) (x + b) = x (x + b) + a (x + b)
                          = x ² + b x + a x + ab
                          = x ² + a x + b x + ab
                          = x ² + (a + b) x + ab

Generally, we call x ² as the first term, (a + b) x as the middle term, and ab as the last term. 

Basic concept: 

a) Step-1: See the sign of the last term.
b) Step-2: Here it is "+" so factorize the product of the coefficient of the first term (here it is 1) and the last term in such a way that the SUM of these two factors must be the coefficient of the middle term.
c) Step 3: Get the factors.

Example-1:

Factorize: x ² + 10 x + 21.

a) Step 1: Here sign of the last term 21 is "+"
b) Step 2: The coefficient of the first term is 1 and the last term is 21, so the product of 1 and 21 is 21. Now the factors of 21 are 3 and 7 and as the sign of the last term is "+", their addition is 3 + 7 = 10 which is the coefficient of the middle term.
                         = x ² + 10 x + 21
                         = x ² + (3 + 7) x + (3 x 7)
                         = x ² + 3 x + 7 x + (3 x 7)
                         = x (x + 3) + 7 (x + 3)
                         = (x + 3) (x + 7)
c) Step-3: So the factors of x ² + 10 x + 21 are (x + 3) and (x + 7) 

Example-2:

Factorize: 8 x ² + 14 x + 5.

a) Step 1: Here sign of the last term 5 is "+"
b) Step 2: The coefficient of the first term is 8 and the last term is 5, so the product of 8 and 5 is 8 X 5. Now the factors of 8 X 5 are 2, 2, 2, and 5 and as the sign of the last term is "+", so, we take two factors in such a way that their sum will be 14. Here 2, 2, 2, and 5 will give us 4 and 10. So, here addition is 4 + 10 = 14 which is the coefficient of the middle term.
                         = 8 x ² + 14 x + 5
                         = 8 x ² + (4 + 10) x + 5
                         = 8 x ² + 4 x + 10 x + 5
                         = 4 x (2 x + 1) + 5 (2 x + 1)
                         = (4 x + 5) (2 x + 1)
c) Step 3: So the factors of 8 x ² + 14 x + 5 are (2 x + 1) and (4 x + 5).

In the next part, we will see the remaining 3 types in detail. These 3 types are given below.
2) (x + a) (x - b) = x ² + (a - b) x - ab
3) (x - a) (x + b) = x ² - (a - b) x - ab
4) (x - a) (x - b) = x ² - (a + b) x + ab  

In the next part, we will see a few examples and some essential formulae.

Click here for the next basics.

 ANIL SATPUTE

97-GRE Math -2- Important Key points and formulas

Click here for the previous basics.

In continuation of part - 1, we will see all the important formulas and useful statements which are to be used in the Math test GRE.

Exponents and Roots:

36) When the same numbers are multiplied several times then the concept of exponents will be used. 5 x 5 x 5 x 5, here we multiplied 5, 4 times so we call 5 as the base and the number 4 is called an exponent. It is written as 5 ⁴ and read as "5 to the power 4" or "5 raised to the 4th power" 5 ⁴ = 5 x 5 x 5 x 5 = 625. 

37) a ^m/n is "a to the power m/n". Here m/n is the power. It means it is the "m-th power of n-th root of a" or we can also say that it is the "n-th root of m-th power of a". It is also denoted as  ⁿ√ a^m 

38) In the expression, a ^1/n, here, it is the n-th root of "a". For any odd value of "n", there is exactly one root for every number "a", even if "a" is negative.
39) In the expression, a^1/n, here, it is the n-th root of "a". For any even value of "n", there are exactly two roots for every positive number "a", and no roots if "a" is negative.

40) Example:  27 has exactly one cube root,  ³√ 27 = 3, but 27 has two 4th roots, ⁴√ 27 and (-⁴√ 27 ), and -27 has only one cube root,  ³√ -27 = -3 but -27 has no 4th root, as it is negative.

Rules of Indices

41) a ^m   x  a ⁿ  = a ^{m + n}
42) a ^m   /  a ⁿ  = a ^{m - n}
43) (a ^m ) ⁿ  = a ^{m x n}
44) a ^o  =  1
45) a ¹  =  a
46) a ^-m  =  1/a ^{m    ,}  ( a ^-1  =  1/a ) , ( 1/a ^-1  =  a )

Operations with Algebraic Expression

47) One or more variables are available in an algebraic expression. Each expression can be expressed as a single term or a sum of two or more terms. Terms of an expression are separated by addition or subtraction.  In other words, two or more terms are separated by a + or - sign.

In an expression, xyz³ + x²yz³ there are two terms that are separated by the + sign whereas in x⁴y³z⁷it is the single term.

Factorization (Factorisation):

a) Factorization of Integers: 

For any integers, we can find the prime factors. Let us find the prime factors of 630.

630 = 63 x 10
       = 3 x 21 x 2 x 5
       = 3 x 3 x 7 x 2 x 5
       = 2 x 3 x 3 x 5 x 7 ( write it in ascending order).

b) Factorization of Polynomial: 

   (x + a) (x + b) = x (x + b) + a (x + b)
                         = x ² + b x + a x + ab
                         = x ² + a x + b x + ab
                         = x ² + (a + b) x + ab
Here the factors of the polynomial  [x ² + (a + b) x + ab], are  
(x + a) and (x + b)

In the next part, we will see a few examples and some essential formulae.

Click here for the next basics.

 ANIL SATPUTE

96-GRE Math-1-Important Key points and formulas

In this blog, we will see all the important formulas and useful statements which are to be used in the Math test of GRE.

1) Integers:

1) Integers are positive numbers, negative numbers, and zero like:{ 0, ±1, ±2, ±3, ±4, … }.

2) The product of two positive integers is a positive integer.

3) The product of two negative integers is a positive integer.

4) The product of one positive integer and one negative integer is a negative integer.

5) 1 is a factor of every integer, but not a multiple of any integer except 1 and - 1.

6) 0 is a multiple of every integer, but not a factor of any integer except 0.

7) The sum of two even integers is an even integer.

8) The sum of two odd integers is an even integer.

9) The sum of an even integer and an odd integer is an odd integer.

10) The product of two even integers is an even integer.

11) The product of two odd integers is an odd integer.

12) The product of an even integer and an odd integer is an even integer.

13) An integer greater than 1 is known as Prime Number if it has only two positive divisors,1 and itself.

14) An integer greater than 1 and not a Prime number then it is called a Composite Number.

Rational Numbers

15) A faction is a number of the form a/b, where a and b are integers and b # 0.  Such numbers are also called Rational Numbers.

16) In the above form a/b, a is known as the Numerator, and b is known as the Denominator.

17) The Numbers which are not Rational Numbers are called Irrational Numbers.

Real Numbers

18) Union of Rational and Irrational Numbers is known as Real numbers.

19) Set of all real numbers can be represented as a Number line known as Real Number Line.

20) Every left-handed number is smaller than a right-handed number.

22) There are four types of intervals depending on their endpoints.

a) a < x < b, it is both open side intervals denoted by (a, b).

b) a ≤ x < b, it is left hand closed and right hand opened interval and is denoted by [a, b).

c) a < x ≤ b, it is left hand opened and right hand closed interval and is denoted by (a, b].

d) a ≤ x ≤ b, it is both sides closed interval and is denoted by [a, b].

23) An absolute value of a number is the distance of that number from 0 on the number line. It is written as |x| and read as absolute of x. It is always positive. 

       |x| = 0 if x = 0,

       |x| = -x if x < 0,

       |x| = x if x > 0,      
24) Properties of Real Numbers:
      a) Commutative Law:
          1) For Addition:  a + b = b + a  Example:  2 + 3 = 5 = 3 + 2
          2) For Multiplication: a * b = b * a   Example:  2 * 3 = 6 = 3 * 2
      b) Associative Law:    
          1) For Addition:  a + (b + c) = (a + b) + c  Example:  2 + (3 + 4)  = 9 = (2 + 3) + 4
          2) For Multiplication: a * (b * c) = (a * b) * c   Example:  2 * (3 * 4) = 24 = (2 * 3) * 4
      c) Distributive Law:    
          1) a * (b + c) = (a * b) + (a * c)  Example:  2 * (3 + 4)  = 14 = (2 * 3) + (2 * 4)

25) " 0 " is an additive identity. This means 0 + a = a, i. e. 0 + 5 = 5.

26) " 1 " is a multiplicative identity. This means 1 * a = a, i. e. 1 * 7 = 7.

27) Additive inverse of any number x is - x. That is the addition of a number and its additive inverse is the additive identity. x + (-x) = 0.

28) Multiplicative inverse of any number x is 1/x. That is the multiplication of a number and its multiplicative inverse is the multiplicative identity. x * (1/x) = 1.

29) Most important note: 0 * a = 0. Any number multiplied by 0 is always 0. One example will definitely be asked on this basic concept. Please remember this example:

" There are 26 alphabets in English. The highest power of x in the expression ( x - a) (x - b) is 2. whereas in (x - a) (x - b) (x - c) is 3. Like this, we used all 26 alphabets in order to say up to z what is the highest power of x ?" 

30) The sum of two positive real numbers is positive.

31) The sum of two negative real numbers is negative.

32) The sum of one positive and one negative number is the difference between them with the sign of the greater number.

33) The product of two positive real numbers is positive.

34) The product of two negative real numbers is positive.

35) The product of one positive and one negative real number is negative.

The next part of this important topic will be published in the next blog.

Click here for the next basics.

ANIL SATPUTE

95-Magic Square-14 (Different view)

Click here for the previous part of the magic square.

For centuries, magic squares have intrigued many minds, and now you can delve into their charm with just a simple input—a date!

With our specially crafted software, you can produce 8 varied magic squares, all centered around the date you provide. What sets this tool apart is that in each of the 8 magic squares, the first row will precisely align with the date you entered (formatted as DD-MM-YYYY).

🔍 What occurs next?

After you input a date:

The software swiftly computes and showcases 8 unique magic square designs.

Every square upholds the mathematical balance of a magic square, where the total of every row, column, and diagonal remains constant.

It’s a fascinating mix of mathematical reasoning and imaginative number play, all beginning with a significant date—be it your birthday, anniversary, or another memorable occasion!

🎯 Why give this a try?

It is an entertaining way to witness math in action.

Ideal for students, educators, and inquisitive individuals

Can be utilized in classrooms or simply for enjoyment

👉 Are you ready to witness the enchantment of numbers?

Click the button below to embark on your journey with a date!

Step-1

🧮📲 Tap the link above to begin your journey. After doing so, you'll be directed to a different page where the real excitement unfolds!

Step-2

🟡 Input any date in the yellow cells using the format “DD MM YY YY” (24 04 20 25).

✨ In just a few moments, the first row of all 8 magic squares will be automatically populated with the date you entered. Simultaneously, the other cells will be filled in, allowing you to quickly view eight distinct and intriguing magic squares, each possessing its unique design!

Step-3

🔗 Refer to the illustration below — on the shown page, there’s a link labeled “Click Here to access all addition patterns.”

You will uncover the concealed mathematical symmetries and addition patterns integrated within each magic square by clicking this link.

When you select this link, a new page will appear, showing all the various kinds of addition patterns present in the magic squares.

Understanding the Arrangement of Magic Square Patterns: A Visual Guide

On this newly unveiled page, you will find a unique and organized layout of eight distinct magic squares. The upper section showcases the first four magic squares, while the lower section presents the last four in a visually appealing manner.

To assist you in visualizing and comprehending the concealed symmetries and numerical relationships, elements from both the upper and lower sections are arranged vertically, one below the other, within specially designed tables. This configuration uncovers various types of addition patterns—from Pattern 11 to Pattern 19.

Each table is a showcase of numerical alignments, including the familiar vertical, horizontal, and diagonal, as well as some innovative combinations that will pique your interest. The diagrams, with their highlighted areas, act as visual aids, showing you exactly where these magic squares are and how their elements contribute to each pattern.

📌 For enhanced clarity, please refer to the illustration provided below.

The Entered Portion Will Appear as Follows:

Once you input the date in the specified yellow cells (in the format DD MM YY YY), the upper section of the page—referred to as the entered portion—will display this date as the first row across all 8 magic squares. This visually confirms that the system has accepted your input correctly.

Here is an example of how this entered portion will look on your screen:

This first row serves as the base for each of the 8 distinct magic squares, which are then automatically populated with numbers that preserve the unique magic square properties and addition patterns.

Special thanks to Abhishek Satpute, Aaswad Satpute, and Jyoti Satpute.

ANIL SATPUTE

94-Math Riddles 01

🧠 Let’s Jump into Some Fun with Riddles! 

Today, we will delve into a handful of straightforward yet thought-provoking riddles. 

They might appear simple at first sight — and they are! 

However, true enjoyment comes from using fundamental mathematical ideas and shifting your perspective. 

🧠 Let’s jump into some enjoyable riddles!

Today, we’re going to take a look at a few straightforward yet intriguing riddles.

They may seem simple, but the fun comes from applying fundamental concepts with a bit of creative thought.

All you need is a curious mind and a new outlook.

Are you prepared to challenge your brain? 😄

We are all acquainted with the decimal number system, which consists of 10 digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Now, here’s the twist...

In this unique riddle, the values of all these digits have been modified!

Indeed, each digit now represents a different, concealed value.

With these altered values, a series of problems has been solved.

Your challenge?

👉 Carefully examine the provided problems.

👉 Uncover the hidden pattern.

👉 Determine the new value associated with each digit.

Get ready — this is more than just a puzzle; it’s a numerical enigma! 🧩✨

Let's discuss some steps in solving such problems.

Solved Example:

Key-1

      The sum of all the digits from 0 to 9 is 58.

Key-2

     853
  + 388
-----------
     818

Key-3

     5381
  + 1573
-------------
    26501

Key-4

     9
  x 6
-----------
   75

Step-1

Using Key 1:

🔍 Let's Discover the Hint!

In the standard decimal system, the total of all digits from 0 to 9 is

0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

However, in our intriguing number puzzle with altered digit values, the total number of digits is 58—something has changed!

Let’s examine it:

🧩 In this puzzle:

The new value assigned to the digit