Number Properties - GMAT

Divisible by means?

For example, 24 is divisible by 4

What happens to a GCD if any of the numbers in a set is 1?

GCD becomes 1

Which one is a higher number? √8 or √12.8

√12.8

How to calculate the mean of the equally spaced set?

(First + Last / 2)

If all the data numbers are the same, what is the Standard Deviation?

0

What is √1 ?

1

The sum of all the factors of a prime number is?

1 + number itself as there are only two factors of a prime number 1 and the number itself

How to find how many trailing zeroes does 24 x 55 x 214 x 135 x 400 have?

1) # of times 10 appears in the product i.e # of times (2 x 5) appear in the product 2) Prime factorize all the numbers separately 3) Add up all the prime factors (in this case it's 2^8 x 5^4 = (2 x 5)^4 x 2^4 So, the answer is 4

Manipulate 0.999996 / 1.002

1) 0.999996 >>> 999996 / 10^6 >>> 10^6 - 4 / 10^6 >>> 1 - 4 x 10^-6 2) 1.002 >>> 10^3 + 2 / 10^3 >>> 1 + 2 x 10^-3 3) 1 - 4 x 10^-6 / 1 + 2 x 10^-3 4) Nominator has to be halved basically and you get 1 - 2 x 10^-3

1) What is the remainder when 1 / 13? 2) Remainder of ¼? 3) Remainder of 2/4?

1) 1 2) 1 3) 2

Difference between the following statements 1) Consecutive prime numbers 2) Consecutive numbers that are prime

1) 2, 3, 5, 7 2) Only 2 and 3

Using negative remainders: Rem 41^31 / 7 ?

1) 41 = 7 x 6 - 1 2) Rem (41 / 7) ^ 31 = -1 ^ 31 = - 1 3) - 1 = - 7 + 6 (Part 1 div. by 7 and Part 2 NOT div. by 7) 4) Remainder is 6

How to check if the number is a prime number?

1) Check if the number is divisible by 2, 3, 5, 7 2) Find the nearest perfect square less than the given number 3) Find the square root of the nearest perfect square less than the given number 4) Check each of the prime factors ≤ the square root of the nearest perfect square less than the given number, for divisibility

If the cyclicity of a units digit = 2, units digit can be found out by the odd or even nature of the exponent

1) Cyclicity of 4 = 2, units digit of 4^62 = units digit of 4^2 = 6, since 62 is even 2) Cyclicity of 4 = 2, units digit of 4^71 = units digit of 4^1 = 4, since 71 is odd

What is the divisibility rule for 11?

1) Let's take an example of 712,596 > (6 + 5 + 1) - (9 + 2 + 7) = 12 - 18 = -6 and therefore it is not divisible by 11 2) Let's take an example of 9,031 > (1 + 0) - (3 + 9) = 1 - 12 = -11 and therefore it is divisible by 11

How to find GCD (Greatest Common Divisor)?

1) Prime factorize each number 2) Take all the common prime factors with their least visible powers 3) Multiply all the terms obtained in Step 2 to get the GCD

How to find LCM (Least Common Multiple)?

1) Prime factorize each number 2) Take all the prime factors from all prime factorizations with their greatest visible powers 3) Multiply all the terms obtained in Step 2 to get the LCM

How do you count the total number of factors for a number?

1) Prime factorize the number 2) (1st power + 1) x (2nd power +1) etc

Find the possible value of x when (x, 18) = 36

1) We have to prime factorize 18 and 36 2) 18 = 2^1 x 3^2 3) 36 = 2^2 x 3^2 4) Given the LCM of 36, we know that it is maxed out so the possibilities for x are the following: for the #2 >>> 0,1,2 and for the #3 >>> 0,1,2 5) We know that 18 has 2^1 but the 36 has 2^2 so x should have the same and then we just calculate possibilities for 3 6) Possibilities are 2^2 * 3*2 = 36, 2^2 x 3^1 = 12, 2^2 x 3^0 = 4

Whare are the factors of 10?

1, 2, 5, 10

If you are given the GCD of two integers, what all can you conclude about the two integers? 1) If the GCD is even, both the numbers are even 2) If the GCD is odd, both the numbers are odd 3) Both the integers have the GCD as a factor 4) If the GCD is 1, both the integers should be equal to 1 5) Number of prime factors in each integer ≥ number of prime factors in their GCD

1, 3, 5 only

What is the remainder when 15^2 is divided by 7?

15 divided by 7 gives us the remainder of 1 so it's the same in a squared form

Which prime numbers are the only consecutive prime numbers?

2 and 3

Simplify 300 / 10^5 = ?

3 / 10^3 Basically dividing by 100

Is 90 a multiple of 30?

30 = 2 x 3 x 5 90 = 2 x 3^2 x 5 It is clear that 90 contains all prime factors of 30 and therefore 90 is a multiple of 30

Every time we multiply two consecutive even numbers the product will always be divisible by ?

8

If A includes all the prime factors of B, what can we about this situation?

A is a multiple of B (90 is a multiple of 30)

How to find how many trailing zeroes does 24 x 55 x 214 x 135 x 400 have?

Break down each number to their primes and eventually you want to see how many 2s and 5s you have so you can add them up and finally you get (2 x 5)^4 x 2^4 and you can say that there are four trailing zeroes

How can you manipulate this equation 0.999999 x 1.000001?

Breaking down 0.999999 first 1) 999999 / 1000000 2) 1000000 - 1 / 10^6 3) 10^6 - 1 / 10^6 4) 10^6 / 10^6 - 1 / 10^6 5) 1 - 10^-6 Breaking down 1.000001 second 1) 100000 + 1 / 10^6 2) 10^6 / 10^6 3) 1 / 10^6 4) 1 + 10^-6

If every prime factor of B is also a prime factor of A, is A divisible by B?

Can not be determined

What are composite numbers?

Composite numbers can be divided by more than itself and the number 1 (not primes). The first seven composite numbers are 4, 6, 8, 9, 10, 12, and 14.

General note: When it says "P is a factor of 10" it means that - The factors of 10 are the numbers that exactly divide 10. You can observe that the numbers 1, 2, 5, and 10 on dividing 10 leaves the remainder as 0. This means 1, 2, 5, and 10 exactly divide the number 10. They are the factors of 10.

Correct

NX will have the same or more number of prime factors than N (Correct/Incorrect)

Correct

Perfect squares CANNOT have odd powers for the prime factors

Correct

Prime factors are let's say 2 x 2 x 2 x 5 here we have 4 prime factors but we have 2 prime numbers. (Correct/Incorrect)

Correct

Remainder of a^n / b = Remainder of (a / b)^n , where a, b, & n are positive integers

Correct

To find remainder when N is divided by D, three basic principles should be used: 1) N = DQ + R 2) 0 ≤ R < D 3) N = Part 1 (divisible by D) + Part 2 ( NOT divisible by D)

Correct

Units digit of 2^(4a+1) = 2 because upon dividing 4a + 1 by 4 (cyclicity of 2), we will always get a remainder of 1 and thus the units digit would be 2^1 = 2

Correct

N = DQ + R 1) DQ is divisible by Divisor 2) R is not divisible by Divisor

Correct 1) 26 = 11 x 2 +4 2) D x Q = 11 x 2 = 22 >>> 22 is divisible by 11 3) R = 4 >>> NOT divisible by 11 Thus, all DIVIDENDS are composed of two parts 1) One divisible by the divisor (DQ) 2) Other NOT divisible by the divisor (R)

Product of "n" consecutive numbers is always divisible by n!, where n! = n x (n-1) x ...1 & n is a positive integer (Correct/Incorrect)

Correct If n = 3, 3! = 3 x 2 x 1 = 6 >>> Product of 3 consecutive numbers 3 x 4 x 5 = divisible by 6 >>> Product of 3 consecutive numbers 16 x 17 x 18 = divisible by 6

Sum of ODD number of consecutive integers is ALWAYS divisible by that odd number (Correct/Incorrect)

Correct Sum of 3 consecutive even integers is divisible by 3

Sum of EVEN number of consecutive integers is NOT divisible by that even number (Correct/Incorrect)

Correct Sum of 4 consecutive even integers is not divisible by 4

If a number has prime factors 3 and 5, is it multiple of 15?

Correct. It will essentially be 15 times something...

Even / Even

Could be either Even or Odd

What are the distinct integers?

Distinct integers mean 2, 4, 6 etc. which are not the same

If you are given choices and asked to calculate what fraction is closest to 1/2, what would be your approach?

Do 1 / 2 - "fraction" >>> whichever has the least fraction is the correct choice because it has the less distance from 1 / 2

When we add/subtract all the elements of a data set with a certain value, the standard deviation or the variance of the data set changes/does not change?

Does not change

Even / Odd

Even (if it's an integer)

A number which is not a perfect square has ___ factors

Even number of factors

What is the GCD of consecutive even integers?

GCD of consecutive even integers has to be 2, since no other even consecutive integer will be a factor of another consecutive integer

How do you count Even or Odd number of factors in a number?

I guess a lot depends on the number 2 because in the odds you assume that as 1 and in evens you take 2's power as the number which will be multiplied to the powers of the other primes as it is in the standard multiplication of the total factors to get the number

If A and B are positive integers, is A / B an integer? i.e is A completely divisible by B?

If every factor of B is also a factor of A, then YES

If two numbers have the same prime factors, the one is a multiple of the other (Correct/Incorrect)

Incorrect (Consider 60 and 90, both numbers have the same prime factors 2, 3, and 5 but neither 60 nor 90 is a multiple of 90 and 60)

Is N a multiple of X? What does this mean in translation?

Is N / X an integer? Is 9 a multiple of 3? Is 9 / 3 an integer?

When we multiply/divide all the elements of a data set with a certain value, the standard deviation or the variance of the data set changes/does not change?

It does change

If two numbers have the same prime factors, what can we say about the numbers?

It does not guarantee that they are the same numbers

If a number is a square of an integer, then...?

It has an odd number of total factors

If a number has odd number of factors, then....?

It is always a perfect square

If a number has just two factors, what does that mean?

It's a prime number

If a number has even number of factors, then....?

It's not a perfect square

All common multiples are multiples of?

LCM (Least Common Multiple) 2 & 3 have LCM of 6 >>> all multiples of 6 have 2 & 3 as their multiples

In the set, if each number is decreased by 5 what happens to the mean?

Mean decreases by 5

In the set, if each number is divided by 5 what happens to the mean?

Mean divides by 5

In the set, if each number is increased by 5 what happens to the mean?

Mean increases by 5

In the set, if each number is multiplied by 5 what happens to the mean?

Mean multiplies by 5

If n^3 is not divisible by 2^3 i.e 8 then what can we infer?

N is not divisible by 2 and N is odd

Does Greatest Common Multiple Exist?

No

Do exponents change the number of factors?

No, N and N^3 will have the same prime factors

If we are given prime factors and then asked to find the value of n, is it possible?

No, it's not possible to find the value. You have to know both the prime factors and their respective powers

If 15 < x < 50, for how many values of x does x/6 have 2 factors?

Note that when it says x/6 has 2 factors it means that the number is prime because it divides on 1 and itself. Here you can also use that approach by dividing 15/6, x/6, 50/6 and you can easily see that there are 3 prime numbers in between

Odd / Odd

Odd (if it's an integer)

All perfect squares have an ___ number of total factors

Odd number of factors

Reminder to memorize the cyclicities of Units Digits

Ok

When can you use the Units Digit method to find a reminder?

Only when we are finding the remainder on division by 2, 5, and 10. Elsewhere, the remainders concept needs to be used

The LCM of (P, 14) = 14; What are the possible values for P?

P = 1, 2, 7 or 14

What happens if in the equation N = DQ + R, R is less than 0?

R should be split into two parts, one divisible by D and other not divisible by D, to get the remainder. 1) 26 = 11 x 3 - 7 2) -7 = -11 + 4 (-11 divisible by 11 and 4 not divisible by 11) 3) 26 = 11 x 3 - 11 + 4 4) 26 = 22 + 4 5) R = 4

What happens if in the equation N = DQ + R, R is more than D?

R should be split into two parts, one divisible by D and other not divisible by D, to get the remainder. 1) 36 = 5 x 4 + 16 2) 16 = 15 + 1 3) 36 = 5 x 4 + 3 x 5 + 1 4) 36 = 5 x 7 + 1 5) R = 1

What's the units digit cyclicity rule for 4?

Regarding 4s, we can say that the units digit of 4^odd = 4 and the units digit of 4^even = 6

Remainder of (a + b) / n ?

Remainder of (a / n) + (b / n) where a & b & n are positive integers

Remainder of (a x b) / n ?

Remainder of (a / n) x (b / n) where a & b & n are positive integers

N and N with a higher power will have ___ ?

Same number of prime factors (but not the same powers)

If you take any set of numbers and find the average value, you will observe that, maximum value ≥ average value ≥ minimum value

So, in statement 1, we are given that average value = minimum value, this is possible only when all the numbers of the set are same that is equal to the minimum value. If at least one number is different, then the average value will be > the least value.

Know how to calculate Standard Deviation

Steps to find the standard deviation are as follows: Step 1: Find the arithmetic mean of the given data set. Step 2: Find the difference between the mean and each element of the data set. Step 3: Square each of the differences. Step 4: Find the average of the squared differences. This is called variance. Step 5: Take the positive square root of the variance.

When can the concept of negative remainders be used?

The concept of negative remainders can be used to find remainders efficiently while dealing with exponents

How do you count numbers between x and y?

Three possible cases here. 1) The number of integers between a and b (b > a) including, can be calculated by b - a + 1 Here including denotes we need to count both a and b. Hence, the number of integers between 3 and 9, including is 9 - 3 + 1 = 7 (which are 3, 4, 5. 6, 7, 8, 9) 2) The number of integers between a and b (b > a) excluding, can be calculated by b - a - 1 Here excluding denotes we cannot count both a and b. Hence, the number of integers between 3 and 9, excluding is 9 - 3 - 1 = 5 (which are 4, 5, 6, 7, :) 3) The number of integers between a and b (b > a) one side including, can be calculated by here one side including denotes we need to count either a or b but not both. Hence, the number of integers between 3 and 9, one side including is 9 - 3 = 6 (which are 3, 4, 5, 6, 7, 8 or 4, 5, 6, 7, 8, 9)

All common divisors are divisors of GCD (Greatest Common Divisor). True/False?

True

For any two integers, LCM x GCD = Product of the two integers

True

GCD of the given numbers can never be grater than any of those given numbers (6, 12, 18 - can't be greater than 6 because it has to be common)

True

GCD of two consecutive numbers is always 1. True/False?

True

In the evenly spaced sets the mean and the median are the same (True/False)

True

LCD or Least Common Divisor is always 1. True/False?

True

Sum of 3 consecutive integers is ALWAYS divisible by 3 (True/False)

True

The LCM (Least Common Multiple) of all the numbers in a set is always greater than or equal to the greatest of those numbers. True/False?

True

The lowest prime factor of a prime number is that number itself. True/False?

True

Units Digit of (75 - 32) = (5 - 2) 3 What's UD (75 - 37) = ?

UD(75 - 37) UD(5 - 7) DOES NOT EQUAL 2 UD(75 - 37) UD(15 - 7) = 8

What happens when the cyclicity of the units digit is 1?

Units Digit remains unchanged

How can you check if the number is divisible by 15? 12? 18?

Use the combination rule 1) 15 can be checked if that number is divisible by both 3 and 5 2) 12 can be checked if that number is divisible by both 3 and 4 3) 18 can be checked if that number is divisible by both 2 and 9

How do you calculate the number of factors? What counts?

We only count distinct prime factors when we are asked to count how many prime factors does the number have. For example 12 has 2 prime factors and not 3

If A and B are positive integers and every factor of B is also a factor of A, is A divisible by B? (A/B)

Yes

If x is divisible by 5, is x a multiple of 5?

Yes

Is 0 a nonpositive number?

Yes

Does the remainder depend on the units digit of the number?

Yes 11^q x 6^pq / 5 >>> 1 x 6 / 5 >>> Remainder is 1 when 6/5

Is the product of three consecutive integers divisible by 3?

Yes, always

What is the formula for the sum of first N natural numbers?

n * (n + 1) / 2