NUMBER PROPERTIES

consecutive integers in a sequence of prime numbers

2 and 3 are the only consecutive integers . We know that between consecutive numbers, one number is even and another is odd. Hence, these are the only consecutive integers in a set of prime numbers where 2 is the only even prime number.

LCM of a set of numbers is divisible by all the numbers in the set

This means LCM is the multiple of a set of numbers

If an expression is divisible by 2 or 4? What does it mean?

This means that the expression is an EVEN number

What are the total number of factors of K = P1^2 * P2^2

Total number of factors of K = (2+1) (2+1) = (3)(3) = 9

If total number of factors of K is 9, what will be the number of prime factors of K?

Total number of factors of K = 9 can be written as a product of numbers in 2 ways. Case 1:- (3)(3) where (2+1)(2+1) => K = P1^2 * P2^2 Case 2:- (9)(1) where )8+1)(0+1) => K = P1^8 * P2^0 so K = P1^8. Therefore the prime factors can be 2 per case 1 or 1 per case 2.

Product of LCM & GCD of two numbers

will always be equal to the product of two numbers themselves

The power of a number doesn't impact the even-odd nature of the number.

(Even)^n -> when n is a positive integer = Even. (Odd)^n = Odd

A perfect square will always have an ODD number of factors

e.g. How many factors does 64 have? Since 64 is a perfect square, total number of factors of 64 will be odd. If the answer choices has only 1 odd number then no need to solve, you have the correct answer. Let's say integer K has prime factors with an even power, then integer K has odd number of factors. So, K must be a perfect square.

nX may have same or more prime factors as X

e.g. If positive integer x has 3 prime factors, how many factors does 4x have?? 4 is 2^2 so if X has a prime factor as 2 included in the total prime factors of 3. Then 4X will have 3 prime factors as well. However, if 2 is not included in the prime factors of X. Then 4X has 4 prime factors.

To find out the remainder when a number is divided by 4, we only need to consider the last two digits of the number.

e.g. for 437986/4 , therefore 86/4 has a remainder of 2.

All prime numbers > 2 can be represented in the form

either 4n + 1 or 4n - 3

First 10 natural numbers

1,2,3,4,5,6,7,8,9,10

Odd / Odd

Odd Integer Always

Odd* Odd is

Odd. The only way a product can be Odd is if every single factor is odd.

Neither Prime # nor Composite number

1 & 0

If X and n are positive integers, then X^n has the same prime factors as X

e.g If positive integer n has 3 prime factors then n^4 has 3 prime factors as well.

(Even)^2 means

(2n)^2 = 4n^2 i.e. divisible by 4

In a list of prime numbers starting from the smallest 2 & 3 are the only consecutive numbers.

2 & 3 are the only consecutive numbers with a difference of 1 between them. For prime #s > 2, difference between any two prime numbers is at least 2. It can be greater than 2 as well.

Any two consecutive positive integers

Any two consecutive positive integers no common factor between them so GCD of any two positive consecutive integers will always be 1.

EVEN INTEGERS

2n (multiples of 2)

Odd Integers

2n + 1

consecutive odd integers

2n + 1, 2n + 3, 2n + 5.....

consecutive even integers

2n, 2n+2, 2n+4.....

What is the rightmost non-zero digit of 90^42?

90^42 = 9^42 * 10^42. Anything with 10^42 will result in 0. (e.g 9^2 *10^2 = 8100) hence to find unit digit of 9^42 we can look @ cyclity of 9^2n = 1, power 42 is evenly divided by 2. So unit digit of 9^42 is 1. Therefore right most non-zero digit of unit digit of 90^42 is 1.

The product of first 100 prime numbers is

ALWAYS EVEN. '2' is the only Even number where as the rest are 99 Odd numbers. Therefore, ExO = E

The sum of first 100 prime numbers is

ALWAYS ODD. '2' is the only even prime # where as the rest are 99 odd prime #s. Therefore, E + O = O

To find Total number of factors

Add 1 to the power of each prime factor and then multiply the resulting numbers

Terms with Even coefficients are

Always EVEN.

A term of the form (even number)* (x) will

Always be Even

Smallest common divisor of 135 & 90

Answer is 1 not 3

HCF/ GCF/ GCD (Greatest Common Divisor) of a given number

Can NEVER be greater than the given number itself. It can also be equal to the given number.

n^3 - n

Can be simplified to n(n^2 -1) where n^2 - 1 can be further simplified into (n+1)(n-1). So there are 3 consecutive integers (n—1)(n)(n+1)

Remainder is always less than

Divisor

Dividend =

Divisor * quotient + remainder

Add or subtract 'LIKES' we get EVEN

E+/- E = E or O +/- O = E

Add or subtract 'UNLIKES' we get ODD

E+/- O = O

When an integer is divided by 2, what will be its remainder?

Each multiple of 2 is an even number. Hence, any integer when divided by 2 will either leave a remainder of 0 or 1. So if an integer with an even last digit will leave a remainder of 0 and an integer with an odd unit digit will leave a remainder of 1.

If an integer is divided by 5, what will be its remainder?

Each multiple of 5 ends in either 5 or 0. So if you know the units digit of an integer that is divided by 5, you can find the remainder. For e.g. an integer having its unit digit as 7 will leave a remainder of (7-5) = 2. An integer having its unit digit as 2 will leave a remainder (2-0) = 2.

Terms with Odd coefficients are

Either Even or Odd. Depending on the variable attached to the coefficient.

Even / Odd

Even Integer

Zero is an

Even Integer neither positive nor negative

In a term of the form (even number) + (x)

Even number plays no role in the even-odd nature of the term. For e.g. Is a+8b+(2n+1)c Even? Here the even term 8b doesn't impact the even-odd nature of this expression. So, the expression will have the same even-odd nature as the sum of {a+(2n+1)c}.

Even / Even

Even or Odd Integer

Even* Even is

Even. As long as there is at least one even factor in a product, the product will be Even. Hence, E*O = E

GCD of a set of numbers divides all the numbers in the set

GCD is the factor/divisor of a set of numbers

Is the number of distinct prime factors of the positive integer X more than 4??

Let's say # of prime factors of X are at least 3. This means # of prime factors of X can be 3,4 or more and hence we don't know for sure of what distinct prime factors of X.

Odd/ Even

NEVER an Integer

Even and Odd numbers can be

Negative

Can Fractions or Decimals be categorized as ODD or EVEN?

No only INTEGERS are Odd or Even

In a term of the form (Odd number)* (X)

Odd number plays no role in the even-odd nature of the term. For e.g. Is a+8b+(2n+1)c Even? Here, in the term (2n+1)c, (2n+1) is an odd number and so plays no role in the even-odd nature of this term. So, the term (2n+1)c will have the same even-odd nature as c. So the entire expression a + (2n+1)c will have the same even-odd nature as the expression a+c

A positive integer can be simplified into

Prime factorization i.e. product of prime #s

If LCM of 36 & X is 72, what is the value of X?

Prime factorizing 72 gives us 2^3*3^2, whereas 36 = 2^2*3^2 (prime factorization of 36). Since 72 has 3^2 in it, X can have either (3^0, 3^1, 3^2). Similarly X will have 2^3 in it as 36 has only 2^2 in it and the LCM has 2^3 in it. Now X can't have any more prime factor other than 2 or 3 as the LCM does not have any other prime factor apart from 2 and 3. So, possible values of X = (2^3*3^0, 2^3*3^1, 2^3*3^2) = (8,24,72)

If Q is a perfect square of an odd positive integer and if 8Q^8 has four prime factors, then how many prime factors does under root Q have??

Q is a perfect square of an odd integer. So let's say Q = N*N and since N is odd Q is odd as well. If Q is odd then prime number 2 is not a factor of Q. 8Q^8 has 4 prime factors which can written as 2^3Q^8. K^a will have same prime factors as K. So 2Q has 4 prime factors. This means {2, prime factors of Q} is 4 hence Q has 3 prime factors. So under root of Q will also have 3 factors.

Decimal part of a decimal quotient is

Remainder / divisor

If K is a factor of positive integer X that has total 8 factors, then how many Prime factors does K^2 X^n have??

Since X has 8 total factors. 8 can be written in 3 ways, 8=2*2*2 or 8=8*1 or 8=4*2 hence X =P1^1* P2^1 *P3^1 or X =P1^7 or X = P1^3* P2^1 so X has either 3 prime factors or only 1 prime factor or 2 prime factors. Per the rule K^2X^n will have the same prime factors as X. As we are not sure about the prime factors of X so prime factors for K^2X^n cannot be determined.

(2^6 x 3^3) - (2^5 x 3)

Take out the common factors i.e. 2^5 x 3(2 x 3^2 - 1) = 2^5 x 3(18-1) = 2^5 x 3 x 17 so the prime factors are 2,3 & 17.

If you pick any two consecutive even numbers we can represent 2k as one of the even number

The next even number is a multiple of 4 for sure. So, we can represent this number as 4K.

If a series of numbers has 2 & 5 and the Q deals with product and units digit

Then Recognize that units digit of the product of this series will always be zero.

If divisor is > dividend

Then integer of quotient = 0 and the remainder equals the dividend.

If a number has 3 factors

Then it must be a square of a prime number

terminating decimal

When a fraction is written in the power of 2 or 5 or both combined then the fraction is said to have a terminating decimal. Note this pattern does not apply if the denominator has any other other number than 2 or 5 or multiple of 2 and 5.

x^ any positive integer is always EVEN

When x is EVEN. So, an even number raised to power does not change its even odd integer.

x ^ any positive integer is always ODD

When x is ODD

If positive integer X has 3 prime factors and 8 total factors, then how many total factors will X^n have where n is a positive integer?

X = P1^a * P2^b * P3^c where (a+1)(b+1)(c+1) = 8. This can be written as (a+1)(b+1)(c+1) = 2*2*2 so a=b=c = 1 so X = P1^1* P2^1 *P3^1. X^n can be written as P1^n * P2^n * P3^n so total number of factors of X^n = (n+1)(n+1)(n+1) i.e. (n+1)^3

X is GCD of 12 & 18 for example can also be referred to/written as

X is HCF of 12 & 18/ X is the highest number which divides both 12 & 18. Focus on the keyword "highest" or "greatest".

Number 13 completely divides X! What does it mean??

X is divisible by 13 / X is a multiple of 13 i.e. X could be any multiple of 13, X =13k

Sum of 3 ODD #s

is Always ODD

Non-negative integers

means 0 and positive integers greater than and equal to 1.

N is a product of integers from 1 to 10

means n = 2*3*4*5*6*7*8*9. Note 1 and 10 are excluded.

A fraction simplifying to an ODD integer is only possible

If both the numerator and the denominator are either even or odd. (Odd/Even can never be simplified to an integer). (Even/Odd if simplified to an integer will always yield an even integer).

Odd/Odd

If simplified into an integer will always yield an Odd integer.

Even/Even

If simplified to an integer will yield an Even or Odd integer.

To check if a number is prime number or not?

You find the square root of that number and see if the number is divisible by the numbers less than the square root of that number.

4n^2 -4n + 1

(2n -1)^2 where 2n-1 is always ODD. An ODD number is represented by 2n-1

What does A + B = Even implies??

A + B = Even implies that either both A & B are even or both A & B are odd.

Every factor of B is a factor of A means:-

It does not mean every factor of A is a factor of B. This only means A is a multiple of B. OR A = kB where A/B = k (k is an Integer)

Every positive integer K can be expressed as

K = P1^m * P2^n * P3^r....... (where P1,P2,P3 are prime factors and m,n,r are non-negative integers)

Any two consecutive positive integers will have a LCM

LCM of any two consecutive positive integers will always be product of the two numbers

If an integer is divided by 10, what will its remainder?

Since each multiple of 10 ends in 0. The integer unit's digit will be the remainder e.g. If 8/10 remainder is 8

If A is the number of factors of a perfect square then A is??

Since the number of factors of a perfect square would always be Odd, then we can deduce that A is odd.

Which of the following cannot be the Greatest common divisor of two positive integers x and y?

Since x & y are given to be positive integers, the sum (x+y) will definitely be greater than the integers x & y and we know that GCD can't be greater than the magnitude of the smallest integer. Hence, x+y cannot be the GCD.

For calculating LCM

Use the highest power of primes of the set of numbers.

For calculating GCD

Use the lowest power of primes of the set of numbers.

Whenever the Q asks about the least possible

You need to figure out LCM

Z is the LCM of 12 & 18 for example can also be referred to/written as

Z is the lowest number which is divisible by both 12 & 18/ Z is the lowest number which has 12 & 18 as its factors. Focus on the keyword "lowest".

A number has 2 & 3 as its prime factors what is the value of that number

We cannot determine the value of the number since we are just given prime factors of that number. We don't know the powers of the prime factors. ( n = 2^a * 3^b)

What is the value of Integer "n" if the integers 2,5 and 7 are the only prime factors of n?

We cannot find out value of "n" unless there is something else given. If n<100 given, then n= 2 * 5 * 7 = 70, otherwise first we need to know the powers. Positive integer n = P1^a * P2^b * P3^c

Sum of Integers from (1 to n)

[n (n+1)] / 2

All prime numbers > 3 can be represented in the form of

either 6n - 1 or 6n + 1. However, we cannot represent prime number 2 & 3 in the form of 6n-1 or 6n +1

Product and Sum of two numbers have the same even/odd nature

only if both are even.