PROPERTIES OF NUMBERS

What is an integer?

A number that can be written without a decimal or a fractional component. Eg: -100, -20, 5, 20 etc.

Is zero positive or negative?

Neither. Zero is neither positive nor negative and has no sign associated with it

What is the sqrt (0)?

0

What is 0³?

0

Is 0 = -0?

True

Zero is a multiple of all numbers. True or false?

True

30⁰ equals ?

1

Is zero even or odd?

Zero is an even number

what are the factors of zero?

Zero. Zero is not a factor of any number except for itself

1 has how many factors?

One. One is the only number with exactly 1 factor

Odd + Odd = ?

Even + even = ?

Odd - Odd = ?

Even - Even = ?

Odd + Even = ?

Even + Odd = ?

Odd - Even = ?

Even - Odd = ?

Odd * Odd = ?

Odd * Even = ?

Even * Even = ?

Even/Odd = ? (Result is an integer)

Odd/Odd = ? (Result is an integer)

Even/Even = ? (Result is an integer)

Odd + Odd = Even

Even + even = Even

Odd - Odd = Even

Even - Even = Even

Odd + Even = Odd

Even + Odd = Odd

Odd - Even = Odd

Even - Odd = Odd

Odd * Odd = Odd

Odd * Even = Even

Even * Even = Even

Even/Odd = Even

Odd/Odd = Even

Even/Even = Even

What is a whole number?

0,1,2,3….

What is a signed number?

Positive or negative numbers are known are signed numbers.

For eg: -2, 2, -4,

Signed numbers don’t include 0

In subtraction, which term is the minuend and subtrahend?

First term in subtraction is known as the minuend and the second term is known as the subtrahend

What is the value of the product or quotient when the signs of both the numbers are positive and/or negative?

When the signs of both the numbers are positive, the product or quotient will also be positive.

When both the numbers have opposite signs, the product or quotient will always be negative

(Non zero base)^even exponent = ?

(Non zero base)^odd exponent = ?

(Non zero base)^even exponent = Always positive

(Non zero base)^odd exponent = If the non zero base is positive then positive. If the non zero base is negative then negative

What are the factors of 16?

1, 2, 4, 8, and 16

What is the first non negative multiple of 2?

0 since 2 × 0 = 0

How many multiples exist for a number?

Infinite

What is a prime number?

A prime number is any integer greater than 1 that has no factors other than 1 and itself

What is a composite number?

A positive number which has more than 2 factors (factors apart from 1 and the number itself)

Which is the only even prime number?

2

What are the prime numbers less than 100?

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

How to determine the total number of factors for an integer? For eg: 2160

• Find the prime factorisation of 2160

  • 2⁴ × 3³ x 5¹

• Add 1 to each exponent and multiply these results. The product will be the total number of factors of 2160

  • (4+1)(3+1)(1+1) = 40

What is the difference between total number of factors vs prime factors for a number? For eg: 20

Prime factors are the factors which cannot be broken down any further. 20 = 2² * 5
For number 20, there are 2 prime factors 2 and 5

Factors are ALL the various numbers that when multiplied together will give the given number. 20 = 4 × 5, 20 × 1, 2 × 10 etc.
For number 20, there are (2+1) * (1+1) = 6 total factors

**There is only one set of prime numbers for a number

What are the number of prime factors vs the total number of unique prime factors for 20?

20 = 2² * 5 (3 prime factors) (2 unique prime factors)

How many prime numbers are there between 1 and 10?

2 3 5 7 = 4 prime numbers

Between doesn’t include the numbers mentioned

How many unique prime factors for 625?

625 = 5⁴ (1 unique prime factor)

25 = 5² (1 unique prime factor)

What is LCM or LCD?

LCM is the least common multiple into which all the numbers in the set will divide.

Find the LCM of 24 and 60?

24 = 3 × 2³

60 = 5 × 3 × 2²

LCM = 3 × 2³ x 5 = 120

How does the LCM compare to the 2 numbers, for which the LCM is requested?

If the 2 numbers share no prime factors, the LCM will be xy

If the 2 numbers share a prime number, the LCM is some number less than xy

What is an HCF?

For a set of positive integers, an HCF is the largest number that will divide into all of the numbers in the set

What is the HCF of 8, 12, and 16?

8 = 2³

12 = 2² × 3

16 = 2⁴

HCF = 2²

What is the HCF of a set of positive integers if they have no prime factor in common?

1

What is the nature of HCF and LCM for a set of positive integers?

• The HCF is always less than or equal to the smallest number in the set. (HCF <= Smallest number in the set)

• The LCM is always greater than or equal to the largest number in the set. (LCM >= Largest number in the set)

For eg: Since 25 divides evenly into 100 then LCM (100,25) = 100 and HCF (100,25) = 25

What is the correlation between HCF and LCM for two positive integers, x and y?

HCF = x * y/LCM

LCM = x * y/HCF

Which of the two - HCF or LCM can be used to solve repeating pattern questions?

LCM can be used to solve repeating pattern questions. It can be used to determine when 2 processes that occur at differing rates or times will co-incide

If x and y are positive integers and x/y is an integer, is x/any factor of y also an integer?

Yes. If x and y are positive integers and x/y is an integer, then x/any factor of y is also an integer.

Since 100/20 is an integer, then factors of 20 are also divisible by 100. This is only applicable for factors of 20 and not multiples of 20. If 100/20 is an integer, then 100/40 may or may not be an integer

If x, y, a, and b are integers, and x ≠ 0, in order for x^a/y^b to be an integer, what can you tell about the nature of a and b?

For x^a/y^b to be an integer, a must be greater than or equal to b. If b > a, a proper fraction will result

If z is divisible by both x and y, should z be divisible by the LCM of x and y?

Yes

If some integer z is divisible by both 3 and 4, z must also be divisible by 12. That is z/12 must be an integer

If z is divisible by x = x/z
If z is divisible by y = y/z

What are the divisibility rules for the following numbers:

• 0

• 1

• 2

• 3

• 4

• 5

• 6

• 7

• 8

• 9

• 10

• 11

• 12

• 0 = No number is divisible by 0

• 1 = All numbers are divisible by 1

• 2 = Ones digit is even

• 3 = Sum of all the digits is divisible by 3

• 4 = Last 2 digits are divisible by 4

• 5 = Last digit is a 0 or 5

• 6 = Number is divisible by 2 and 3

• 7 = Just do the division

• 8 = Number is even + last 3 digits are divisible by 8

• 9 = Sum of all digits divisible by 9

• 10 = Last digit is 0

• 11 = Sum of odd numbered places - Sum of even numbered places is divisible by 11

• 12 = Number divisible by 3 and 4

Product of n consecutive integers is always divisible by?

Product of n consecutive integers is always divisible by n! and all factors of n!

Product of 3 consecutive numbers must be divisible by 3!

Product of n consecutive even integers is divisible by?

Product of n consecutive even integers is divisible by 2ⁿ * n!

For eg: Product of 2 consecutive even integers will always be divisible by 2² * 2! = 8

Product of n consecutive odd integers is divisible by?

There is no special divisibility rule for the product of n consecutive odd integers. The rule only works for even integers.

What is the general formula for division?

x/y = Q + r/y

x = Numerator

y = Denominator

Q = Quotient (must be an integer)

r = Remainder (must be non negative and less than denominator y)

**In some situations, the quotient will be 0. This is perfectly fine!

When the division of 2 integers results in a decimal, what can we infer about the remainder. For eg: What can be infer about the remainder when x/y=9.48?

TL;DR:

• When the division of 2 integers results in a decimal, the remainder must be a multiple of the most reduced fractional remainder.

• 0.48 can be simplified to 48/100 = 12/25. We can infer that the actual remainder is a multiple of 25

LONG:
When dividing two integers x and y, the result can be expressed as:

x/y = q + r/y

Here, q is the integer quotient, and r is the remainder, which satisfies 0 ≤ r < y

Given x/y = 9.48, we can infer:

1. Integer Quotient (q): The integer part of 9.48 is 9. Therefore, q=9.

2. Fractional Part: The decimal part, 0.48, represents the fraction of the divisor y that makes up the remainder r.

x/y = 9 + r/y {r/y = 0.48}

To find the remainder r:

r/y = 0.48 =>  r = 0.48 * y

Since r must be an integer, y should be chosen such that 0.48 * y results in an integer.

Determining y:

The decimal 0.48 can be expressed as the fraction 12/25 * y. Therefore:

0.48 = 12/25; r/y = 12/25; r = 12/25 * y

For r to be an integer, y must be a multiple of 25.

How can you convert a decimal remainder into an exact integer remainder? For eg: 9/5 = 1.8, determine what the integer remainder is?

To determine what the integer remainder is, we can multiply the decimal component of the result of the division (0.8) by the divisor (5) to get 4. The remainder is 4.

**To use this strategy accurately, we must know the denominator

How can we correct excess remainder?

An excess remainder is a remainder which is greater than the divisor. We can remove excess remainders by subtracting the divisor from the remainder till the remainder is less than the divisor. If the remainder is less than 0, we must correct the remainder by adding the divisor

What is the remainder for (12 × 13 × 17)/5?

12/5 = 2 2/5; R = 0.4 × 5 = 2

13/5 = 2 3/5; R = 0.6 × 5 = 3

17/5 = 3 2/5; R = 0.4 × 5 = 2

We can multiply the remainders => 2 × 3 × 2 = 12. since 12 is greater than 5, we must remove an excess remainder

12 - 5 = 7

7 - 5 = 2

2 is less than 5, so the remainder is 2

**Remainders can be multiplied, added, subtracted as long as the excess remainders can be taken care of

**If remainder == divisor, still need to remove the excess remainder

What are trailing zeros and how do we calculate the number of trailing zeros in a given number?

Trailing zeros are the number of zeros at the end of the number. Trailing zeros are created by (5×2) pair. Each 5×2 pair creates one trailing zero. The number of 5×2 pairs in the prime factorisation of a number gives us the number of trailing zeros

If we have more factors of 5 over 2s or vice versa - the limiting factor is the number of 5s and 2s that are present, whichever is fewer.

How many trailing zeros are present in 5³⁰⁰ × 2²⁹⁸

298 trailing zeros as we can match the 298 2s with 5s

Calculate the number of trailing zeros in 50⁸ × 8³ × 11²? Also calculate the number of digits in the resulting integer

• Prime factorise the number - (5^16 × 2^16) x 11^2

• 16 trailing zeroes * 11^2 × 2

• 242 × 16 trailing zeroes = 19 digits

What are leading zeros in a decimal?

Leading zeros are zeros to the right of the decimal but before the first non zero number.

0.2 has no leading zeros

0.02 has 1 leading zeros

0.002 has 2 leading zeros

Calculate the number of leading zeros in the following:

• 1/8

• 1/80

• 1/800

In all the questions, we notice that the denominator is not a perfect power of 10 (Perfect Power of 10: Equal number of 2s and 5s), so the number of leading zeros will be k-1 (k = number of digits)

1/8 = Single digit = No leading zeros

1/80 = Double digit = One leading zeros

1/800 = Three digit = Two leading zeros

Calculate the number of leading zeros in the following:

• 1/10

• 1/100

• 1/1000

Leading zeros: Number of zeros that appear immediately after the decimal point before any non zero digits

Answers to the questions:

• 1/10 = 2 digits = No leading zeros

• 1/100 = 3 digits = 1 leading zero

• 1/1000 = 4 digits = 2 leading zeros

Explanation:

• For fractions of the form 1/n, where n is a positive integer:

  • If n is a power of 10 (e.g., 10, 100, 1000), the decimal representation is 0.0…01, with the number of leading zeros equal to one less than the power of 10.

    • Example: 1/100 = 0.01 has one leading zero.

  • For other values of n, the number of leading zeros depends on the position of the first non-zero digit in the decimal expansion.

Alternative Approach:

To estimate the number of leading zeros without full decimal expansion:

1. Compute the logarithm: Calculate -log₁₀(n)

2. Determine leading zeros: The integer part of -log₁₀(n) gives the number of leading zeros.

Example:

• For 1/500:

  1. Compute logarithm: −log⁡₁₀(500)≈−2.699

  2. Leading zeros: The integer part is 2, indicating two leading zeros

Product of any set of n consecutive integers is divisible by all integers between 1 and n inclusive? True or False

True

What is the largest possible integer value of n such that 21!/3^n is an integer?

21/3^1 = 7

21/3² = 2

21/3³ = 0

7 + 2 + 0 = 9 threes in 21!, and thus the largest possible integer value of n is 9

What is the largest value of n such that 40!/6^n is an integer?

= 40!/2^n * 3^n

We could find the total number of 3s, then determine how many 2 × 3 pairs we can generate. However, since there are fewer 3s than 2s in 40!, the number of threes will limit the number of (2×3) pairs in 40! Thus, we can determine the number of pairs just by determining the number of 3s

40/3^1 = 13

40/3² = 4

40/3³ = 1

40/3^4 = 0

18 number of 2×3 pairs

What is the largest integer value of n such that 30!/4^n is an integer?

30!/4^n = 30!/2^2n

30/2^1 = 15

30/2² = 7

30/2³ = 3

30/2^4 =1

30/2^5 = 0

26 2s in 30!. Now we need to determine the maximum value of n such that 30! is an integer

2^26/2^2n => 26 >= 2n for the fraction to result in an integer

Therefore, n <= 13. The largest value of n must be 13

What is a perfect square?

A perfect square, other than 0 and 1, is a number such that all of it’s prime factors have even exponents

For eg: 2^4 (4 is divisible by 2)

2² (2 is divisible by 2)

Perfect squares always end in?

A perfect square must always end in 0,1,4,5,6, and 9

A perfect square can never end in?

2,3,7, or 8

What is a perfect cube?

A perfect cube, other than 0 and 1, is a number such that all of it’s prime factors have exponents that are divisible by 3

For eg: 64 = 4³ (3 is divisible by 3)

2^6 (6 is a multiple of 3)

How can we tell whether the given fraction will terminate?

The decimal equivalent of the fraction will terminate iff the denominator of the reduced fraction has a prime factorisation that contains only 2s, 5s, or both.

If the prime factorisation of the reduced fractions contains anything other than 2s or 5s, the decimal equivalent will not terminate

Do divisors exhibit remainder patterns?

Yes, all divisors exhibit remainder patterns.
For eg: consecutive division by 7 will follow the repeated pattern 0,1,2,3,4,5,6,0,1,2,3

When positive integer Q is divided by 11, the remainder is 9. What is the remainder when Q + 5 is divided by 11?

Q / 11 => R = 9

Q+1 / 11 => R = 10

Q+2 / 11 => R = 0

Q+3 / 11 => R = 1

Q+4 / 11 => R = 2

Q+5 / 11 => R = 3

What is the remainder when 2^62 is divided by 7?

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

2^5 = 32

.

.

2^60 = __6

2^61 = __2

2^62=___4

Remainder = 4

Is it true that when a positive number is raised to consecutive powers, that are positive integers (both the base and exponent are positive integers), a pattern will arise in the units digit of the product?

Yes.

**The units digit pattern of the powers of any integer greater than 9 has the same units digit pattern as the powers of it’s units digit. For eg: The powers of 12 have the same units digit pattern as the powers of 2

What is the remainder when a whole number is divided by 10 and 100?

Whole number/10 = Remainder is the units digit of the dividend

Whole number/100 = Remainder is the last 2 digits of the dividend

Is the remainder constant when integers with the same units digit are divided by a positive integer?

Yes. For eg: 9/5 = 1 4/5 (R = 4)

19/5 = 3 4/5 (R = 4)

What is an evenly spaced set?

In an evenly spaced set, the numbers in the set increase by the same amount and therefore share a common difference

Will 2 consecutive integers share the same prime factors?

No. 2 consecutive integers will never share the same prime factors

What is the HCF of 2 consecutive integers?

1

From a number of fractions given in the question, how can I check which fraction is the biggest or smallest?

Method 1: Cross-Multiplication

For comparing two fractions a/b​ and c/d​:

1. Cross-multiply to compare: Calculate a x d and c x b

2. Compare the results:

   • If a x d > c x b, then a/b > c/d​

   • If a x d < c x b, then a/b < c/d​

Method 2: Decimal Approximation

Convert each fraction to a decimal by performing the division. The fraction with the largest decimal value is the largest fraction and vice versa

Method 3: Common Denominator

Find a common denominator for all the fractions and then compare the numerators. This method can be cumbersome for large fractions but is conceptually straightforward.

Method 4: Approximation and Estimation

If the fractions are very large, approximate them to a simpler form or use known properties of fractions. For example, knowing the fractions are close to known values like 0.5, 0.75, etc., can help quickly estimate which is larger.

• Eg: Compare which one is greatest amongst 125/256, 320/625, 455/841, 912/1600, 789/1500

  • Comparing 125/256, 320/625

    • 125 × 625 < 320 × 256

    • 125/256 < 320/625

  • Comparing 320/625, 455/841

    • 320 × 841 < 625 × 455

    • 320/625 < 455/841

  • Comparing 455/841, 912/1600

    • 455 × 1600 < 841 x 912

    • 455/841 < 912/1600

  • Comparing 912/1600, 789/1500

    • 912 × 1500 > 789 1600

    • 912/1600 > 789/1500

• Therefore 912/1600 is the greatest fraction of all

How to find the value of positive integer N, if positive integer N divided by positive integer d leaves remainder r?

If positive integer N divided by positive integer d leaves remainder r, then the possible values of N are: r, r + d, r + 2d etc.

Eg: If positive integer N divided by 15 leaves a remainder of 7, what are the 5 smallest possible values of N?

1 = r = 7

2 = r + d = 22

3 = r + 2d = 37

4 = r + 3d = 52

5 = r + 4d = 67

How to find the value of positive integer N, if positive integer N divided by positive integer j leaves remainder b, and if positive integer N divided by positive integer k leaves a remainder c

• Find the smallest possible value of N (List the first few values that satisfy each remainder statement until you spot the same value in each list)

• Add the LCM of j and k to this smallest value as many times as necessary

• For eg: When positive integer N divided by positive integer 6 leaves remainder 2, and if positive integer N divided by positive integer 8 leaves a remainder 4

  • Step 1: Find the smallest value of N

    • N divided by positive integer 6 leaves remainder 2 = 2,8,14,20,26

    • N divided by positive integer 8 leaves a remainder 4 = 4,12,20

  • Step 2: Add the LCM of j and k as many times as necessary

    • LCM = 24

    • 20,44,68,92..

What are the answers to the following values:

• log₁₀(2)

• log₁₀(3)

• log₁₀(5)

• log₁₀(7)

• log₁₀(10)

• log₁₀(a * b)

• log₁₀(a/b)

• log₁₀(a^b)

• log₁₀(50)

• log₁₀(18)

• log₁₀(450)

• log₁₀(2) = 0.301

• log₁₀(3) = 0.477

• log₁₀(5) = 0.699

• log₁₀(7) = 0.845

• log₁₀(10) = 1

• log₁₀(a * b) = log₁₀(a) + log₁₀(b)

• log₁₀(a/b) = log₁₀(a) - log₁₀(b)

• log₁₀(a^b) = b * log₁₀(a)

• log₁₀(50) =

  • 50 = 5 × 10

  • log₁₀(50) = log₁₀(5 × 10)

  • log₁₀(50) = log₁₀(5) + log₁₀(10) => 0.699 + 1 => 1.699

• log₁₀(18) =

  • 18 = 2 × 3²

  • log₁₀(18) = log₁₀(2 × 3²)

  • log₁₀(50) = log₁₀(2) + 2 log₁₀(3) => 0.301 + 2 * 0.477 => 1.255

• log₁₀(450) =

  • 450 = 45 × 10 = 3² × 5² × 2

  • log₁₀(450) = log₁₀(3² × 5² × 2)

  • log₁₀(50) = 2 * log₁₀(3) + 2 * log₁₀(5) + log₁₀(2) => 2 × 0.477 + 2 × 0.699 + 0.301 => 2.653

In a decimal number, a bar over one or more consecutive digits means that the pattern of digits under the bar repeats without end. What fraction is equal to 7.583¯ ?

A. 91/3

B. 91/4

C. 91/6

D. 91/9

E. 91/12

When a decimal can be converted to a fraction, it’s either a finite decimal (e.g., 1.2, 3.45) or a repeating decimal (e.g., 0.666..., 0.07878…). We know how to convert a finite decimal to fraction (e.g., 1.2 = 12/10 and 3.45 = 345/100). We can do the same for a repeating decimal, but instead of “over 10”, “over 100,” it’s “over 9”, “over 99” and so on. Of course, that depends on the number of digits in the “block of digits” that are repeating. For example, in 0.666…, the block of the digits that is repeating is only 6. So the block is 6 and there is only 1 digit repeating. To convert the decimal to a fraction, we put the block over the number the number 9 and we have 0.666... = 6/9 = 2/3. Let’s have another example, in 0.375375...., the block is 375 and the number of digits repeating in this block is 3. Therefore, we are going to put 375 over 999. In other words, 0.375375… = 375/999. However, this method is only true when the block that is repeating is right after the decimal point. It will not be true if the decimal is 0.07878… or in our example, 7.58333…. On the other hand, we can modify our rule to make it work for decimals such as 0.07878… and 0.00375375…, i.e., decimals with all zeros (including the one to the left of the decimal point) before the repeating block. We will tweak the rule as follows:

Again count the number of digits in the block that are repeating (let’s say it’s m) and also count the number of zeros between the decimal point and the block (let’s say it’s n). Then the denominator of the fraction (before reducing) is the integer with m nines followed by n zeros. For example, if m = 2 and n = 1, put the block over 990. If m = 3 and n = 2, put the block over 99900 and so on. Therefore, 0.07878… = 78/990 and 0.00375375… = 375/99900.

At this point, you might say we still haven’t answered how to convert the decimal 7.58333… to a fraction since the block that is repeating, 3, is not preceded by 0s. Yes, it’s true, however, for such a decimal, we can always express it as a sum of a finite decimal and a repeating decimal that we’ve discussed. Notice that 7.58333… = 7.58 + 0.00333… Therefore, we have:

7.58333… = 7.58 + 0.00333… = 758/100 + 3/900 = 758/100 + 1/300 = 2274/300 + 1/300 = 2275/300 = 91/12

Answer: E

If n is a positive integer and the product of all integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?
A. 10
B. 11
C. 12
D. 13
E. 14

We are told that
n!=990∗k=2∗5∗32∗11∗k
n!=990∗k=2∗5∗32∗11∗k --> n!=2∗5∗32∗11∗k
n!=2∗5∗32∗11∗k which means that n! must have all factors of 990 to be the multiple of 990, hence must have 11 too, so the least value of n is 11 (notice that 11! will have all other factors of 990 as well, otherwise the least value of n would have been larger)..

Answer: B.