1) Types of numbers




Note \(\sqrt{negative numbers}\) or \(\frac{number}{0-zero}\) are NOT real !!

On the GRE by default, any number mentioned in the quant section is a real number so must NOT assume that it is an integer unless stated clearly




2) Number Line








3) Basic operations on numbers










4) Place and Face Value

Place Value	Face Value
Place value is the value represented by a digit in a number according to its position in the number.	Face value is the actual value of a digit in a number.
To get the place value of a number, we multiply the digit value with its numerical value. For example, in the number 452, the place value of 5 is (5 × 10) = 50, since 5 is in tens place.	The face value of a digit is the number itself. For example, in the number 452, the face value of 4 is 4.
The place value of a number depends upon the position of the digit in the number.	The face value is independent of the position of the digit in the number.
The place value of a digit in ones place is always a single digit and the place value of every next digit to the left increases by one more digit.	The face value of a number is always a single digit.

5) Basic operation on Even and Odd Numbers

Basic operation on Even and Odd Numbers
Math operations	Application	Examples
Addition/Subtraction	Even \(\pm\) Even = Even; Odd \(\pm\) Odd = Even; Even \(\pm\) Odd = Odd Any number of even numbers can make an even number; but only an even number of odd numbers can make an even number.	4+2=6;4-2=2; Even 7+3=10;3 -7=-4; Even 5+2=7;5-2=3; Odd
Multiplication	Even \(\times\) Even = Even; Odd \(\times\) Odd = Odd; Even \(\times\) Odd = Even Multiplication of any number of odd numbers always results in another odd number. If the product of two integers is Even and one of them is an Even integer, we cannot assess on the nature of the other number; i.e. if it is Even or Odd	4 \(\times\) 2=8; Even; 3 \(\times\) 3 = 9; Odd; 2 \(\times\) 3 = 6; Even
Division	Not every operation of Division is completely divisible. The division could leave remainder or not Odd / Odd = Odd; Even / Odd = Even; Odd / Even = Not divisible; Even / Even = Even or Odd	9/3=3; Odd 10/5=2; Even 11/2=? Not divisible 4/2=2; Even 10/2=5; Odd

6) Prime Numbers

A number greater than 1 which has no factor other than 1 and the number itself is a Prime Number
A number with two distinct factors.
We do have infinite prime numbers and apparently, to our knowledge today, there is NO clear pattern in their unfolding toward the infinite on the number line.
Prime numbers are only positive.
Two (2) is the oNLy positive Even prime numbers
The difference between any two primes greater than 2 is always even.
Other than 2 and 3, if divided by 6, will always leave a remainder of 1 or 5. Any prime cab be written or expressed as \(6k+1\) where k is some positive integer.
Not all numbers, however, which leave a remainder of 1 or 5 when divided by 6 are prime such as 25,35.


Verifying if a number integer and positive is a prime, regardless

1) Take square root of the number, if it is an integer then the number is NOT a prime number;

2) Consider the largest integer closest to the square root of the number n is m , where \(m < \sqrt{n}\)

3) Divide n by all the prime numbers from \(2 to m\), inclusive: if n is divisible by any number among them, the number n is NOT a prime, else is a prime.

\(n=101\)

\(10 < \sqrt{101} <11\) is not an integer

\(101 = \approx 10=m\)

Divide 101 by prime numbers 2 to 10 or 2,3,5,7. Since 101 is not divisible by any prime number listed then 101 MUST be a prime number.










7) Rounding




Rounding Rules

more than 5 > up
less than 5 < down

~ for decimal fractions

1. Look at the digit in the place value to be rounded to.
2. Increase it by 1 if the digit to the right of it is 5 or more.
3. Leave it the same if the digit to the right of it is less than 5.
4. Remove everything to the right of the digit.

Example:

Round 625.475 to 1 decimal place: 625.475 ≈ 625.5
~ when rounding to 10 or above there's an important change to step 4.
4. Replace whole numbers to the right of the digit with zero(s), then remove everything to their right.

Examples:
Round 625.475 to 10: 625.475 ≈ 630
Round 625.475 to 100: 625.475 ≈ 600



8) P.E.M.D.A.S






9) Divisibility Rules








10) Factors

Factor

• a whole number that divides exactly into another number.
• a whole number that multiplies with another number to make a third number.

Factors Pair


• a pair of numbers multiplied together form another number called their product.





Number of factors


A number N can be written as \( N=a^xb^yc^z\), where a,b,and c ar prime factors of the number N, and x,y, and z are positive integers.


Number of factors : (including 1 and the number itself)\(=(x+1)(y+1)(z+1)\);

Factors of \(24=2*2*2*3=2^3*3^1\)

The exponent are 3 and 1 ====> \((3+1)(1+1)=2*4=8\)


Sum of all the factors


A number N can be written as \( N=a^xb^yc^z\), where a,b,and c ar prime factors of the number N, and x,y, and z are positive integers.

Sum of all the factors (including 1 and the number itself)

\(\left[ \begin{array}{cc|r} \frac{a^{x+1}-1}{a-1} \end{array} \right]\) \(\left[ \begin{array}{cc|r} \frac{b^{y+1}-1}{b-1} \end{array} \right]\) \(\left[ \begin{array}{cc|r} \frac{c^{z+1}-1}{c-1} \end{array} \right]\)



11) Perfect Square Numbers

A perfect square number is a non-negative integer that can be expressed as the product of an integer with itself. I.E., a number that is a square of any integer is called a Perfect Square number. Perfect square numbers are always non-negative.

Unit's digit	0	1	4	5	6	9
Ten's digit	0	even	even	2	odd	even

Some properties of a perfect square:

• Perfcet square number ALWAYS has even number of powers of prime factors;
• The number of distinct factors of a perfect square number is ALWAYS odd. (By factors we refer to only the "positive numbers that divide the number withouth remainder);
• The sum of distinct factors of a perfect square number is ALWAYS odd;
• A percet square number ALWAYS has an odd number of odd-factors, and even number of even-factors;

Example
Of the following, which one is a perfect square?

a) 126736
b) 137826
c) 149913

The only number that has both of the combinations at the same time is a) 126736

The unit digit is 6, AND the tenth digit is ODD

Question to practice about perfect square

https://gre.myprepclub.com/forum/if-a-perf ... tml#p59618
https://gre.myprepclub.com/forum/a-perfect ... tml#p32573
https://gre.myprepclub.com/forum/positive- ... tml#p39436
https://gre.myprepclub.com/forum/a-perfect ... tml#p51029



12) Multiples






13) HCF (GCD / GCF) & LCM OF NUMBERS



HCF : It is the greatest factor common to two or more given numbers. It is also called GCF OR GCD (greatest common factor or greatest common divisor). e.g. HCF of 10 & 15 = 5, HCF of 55 & 200 = 5, HCF of 64 & 36 = 4

To find the HCF of given numbers, resolve the numbers into their prime factors and then pick the common term(s) from them and multiply them if more than one. This is the required HCF.

LCM : Lowest common multiple of two or more numbers is the smallest number which is exactly divisible by all of them. e.g. LCM of 5, 7, 10 = 70, LCM of 2, 4, 5 = 20, LCM of 11, 10, 3 = 330

To find the LCM resolve all the numbers into their prime factors and then pick all the quantities (prime factors) but not more than once and multiply them. This is the LCM.

NOTE:
1. LCM x HCF = Product of two numbers (valid only for “two”)
2. HCF of fractions = HCF of numerators ÷ LCM of denominators
3. LCM of fractions = LCM of numerators ÷ HCF of denominators


Calculating LCM :

Method 1 : Factorization Method

Rule – After expressing the numbers in terms of prime factors, the LCM is the product of highest powers of all factors.

Q. Find the LCM of 40, 120, 380.

A. 4\(0 = 4 \times 10 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5^1\)

\(120 = 4 \times 30 = 2 \times 2 \times 2 \times 5 \times 3 = 2^3 \times 5^1 \times 3^1\)

\(380 = 2 \times 190 = 2 \times 2 \times 95 = 2 \times 2 \times 5 \times 19 = 2^2 \times 5^1 \times 19^1\)

⇒ Required LCM = \(2^3 \times 5^1 \times 3^1 \times 19^1 = 2280\).

Method 2 : Division method

Q. Find the LCM of 6, 10, 15, 24, 39




Ans. LCM = 2 x 2 x 3 x 5 x 2 x 13 = 1560.


Calculating HCF


Factorization
After expressing the numbers in term of the prime factors, the HCF is product of COMMON factors.

Ex. Find HCF of 88, 24, 124

\(88 = 2 \times 44 = 2 \times 2 \times 22 = 2 \times 2 \times 2 \times 11 = 2^3 \times 11^1\)
\(24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1\)
\(124 = 2 \times 62 = 2 \times 2^1 \times 31^1 = 2^2 \times 31^1 ⇒ HCF = 2^2\)