• “We all use math every day; to predict weather, to tell time, to handle money. Math is more than formulas or equations; it's logic, its rationality, it’s using our mind to solve the biggest mysteries we know” – Numb3rs

We started dealing with numbers early in our childhood, but still this area poses a major challenge to majority of us preparing for aptitude tests. We either overlook this topic presuming it to be simple or are intimidated by the concepts related with it. But the truth is that anyone who intends to prepare seriously for an aptitude test is left with no other option but to study number systems top to bottom. Reasons being,

• A big share of questions in Quant comes from Number systems. Once you are comfortable with Number system concepts, the so called herculean task of clearing Quant cutoff shall become much easier.
• Mastery in Number system concepts will give you an extra edge in two very important and calculation intensive sections - Quant and DI.
• Your comfort level with numbers will reflect in your solving speed, saves you time and hence boost your overall score.
• The Concepts handled in Number systems are well known to us and all that is required is a brush up.
• Number system is a major error-prone area. Mistakes not just take away our hard earned marks but also waste precious time.

Let’s make peace with numbers and as the adage goes "A friend in need is a friend indeed”. This friendship will come in very handy during our exams!!

Natural numbers

{1, 2, 3, 4…}
Natural numbers have 2 main purposes
counting (1, 2, 3, etc.)
ordering (1st, 2nd, 3rd, etc.)

Whole numbers

{0, 1, 2, 3…}
Why Zero is not a natural number ?

Integers

{…-2, -1, 0, 1, 2…}

Integer is the Latin for "Whole". It denotes positive, zero and negative.

Decimals

There are 3 types of decimal numbers

• Terminating decimal - has a finite decimal value. (e.g. 10.243 )
• Infinite recurring decimals - will not terminate and has a recurring pattern in it. (e.g.: 4.333333333333333…. , where recurring part is 3)
• Infinite non recurring decimals - will not terminate and has no recurring pattern in it (e.g.: 3.453298098736253322….)

Fractions

A common fraction (also known as a vulgar fraction or simple fraction) is a rational number written as a/b where the integers a & b are called the numerator and the denominator, respectively.

Proper fraction: numerator is smaller than the denominator
absolute value of proper fraction is less than one - fraction is between -1 and 1

Improper fraction (also called Top-Heavy fraction) :
numerator is larger than or equal to denominator
absolute value of improper fraction is greater than or equal to 1

Mixed fraction: sum of a non-zero integer and a proper fraction

To convert a mixed number into an improper fraction, multiply the whole number by the denominator and add it to the numerator. This becomes the numerator of the improper fraction; the denominator of the new fraction is the same as the original denominator.

Even numbers

Any integer that can be represented in the form 2n, where n is an integer

Odd numbers

Any integer that can be represented as 2n + 1 or 2n – 1, where n is an integer

Note that even and odd numbers are always integers.

Positive/Negative

Positive: Numbers greater than zero

Negative: numbers less than zero

Real numbers

Either rational (5, -2, 4/3 …) or irrational (√2, ∏ …)

Either algebraic or transcendental

Either positive, negative, or zero

Every real number can be represented on the number line.

Rational numbers

Numbers which can be represented in the form of p/q, q # 0

2, -2, 2.2, 2/3, 4.333333… (39/9) are all rational numbers.

Any infinite decimal values with recurring pattern can be represented in p/q form.

Consider 0.33333333…  This can be converted to p/q form as 3/9, which is the recurring part divided by n nines where n is the number of digits in the recurring part (here 1).
We can write, 0.464646… = 46/99; 2.383383383… = 2 + 383/999

Now what if the number is 0.34583838383… where recurring pattern (83) starts after some digits and such numbers are called impure recurring decimals.Here method to get the p/q form is to subtract the non recurring part (345) from the number formed by non recurring digits and the recurring pattern written once which will give us the numerator (p). For denominator (q) write as many nines as the number of digits in the recurring part (here 83 is the recurring part, so 99) followed by as many zeros as the number of digits in the non recurring part (here 345, hence three zeros).  We get denominator as 99000.

0.34583838383… = (34238 – 345) / 99000 = 34238/99000

Irrational numbers

Any number that cannot be written in p/q form.

Infinite non recurring decimal values where there is no end and no recurring pattern are irrational numbers. Examples are √2, √3, √5, etc…

Prime numbers

A natural number greater than unity is a prime number if it does not have a factor except for itself and unity.

E.g.: 2,3,13,29,53,79 etc…

A simple method of verifying the primality of a given number n is known as trial division. It consists of checking whether n is divisible by any prime number which is less than √n. If not, then n is a prime number.

For 29, √29 is between 5 and 6. Prime numbers less than 6 are 2, 3 and 5. Number 29 is not divisible by any of these numbers making 29 a prime number.

Prime numbers of the form 2p−1 are Mersenne primes (P – Prime number)

A prime number p is a Sophie Germain prime if (2p+1) is also a prime.

Prime numbers whose difference is 2 are called twin primes. e.g.: (3, 5), (11,13)

Prime numbers less than 1000 (168 prime numbers) are

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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997.

Composite numbers

A composite number is a positive integer that has at least one positive divisor other than one or itself.

One is neither prime nor composite.

Every composite number can be written as the product of two or more primes.
E.g. 100 = 22­* 52

A composite number with two prime factors is a semi prime.

A composite number with three distinct prime factors is a sphenic number.

If none of its prime factors are repeated, it is called square free.

Powerful number

A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m.

A powerful number can be written as the product of a square and a cube, that is, m = a2b3 (a, b are positive integers).

E.g. 1, 4, 8, 9, 16, 25, 27…

ZERO

Zero is neither positive nor negative.

Indian mathematician Brahmagupta's ‘Brahmasphuasiddhanta’ is the first book that mentions zero as a number; hence Brahmagupta is considered the first to formulate the concept of zero.

Zero fits the definition of even number: it is an integer multiple of 2, namely 0 × 2.

Zero is not a natural number as one normally does not start counting with zero. Yet zero does represent a counting concept: the absence of any objects in a set. To resolve this issue, some mathematicians define the natural numbers as the positive integers.

Zero is neither a prime number nor a composite number.

π (PI)

The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter, and is approximately equal to 3.14159.

π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (22/7 is  commonly used to approximate π)

π is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients.

The digits of π have no apparent pattern

e (Napier's constant)

The number e is approximately equal to 2.71828

e is the base of the natural logarithm.

e is irrational

e is transcendental

Perfect Squares

An integer that is the square of an integer.

Squares are always non negative.

A positive integer that has no perfect square divisors except 1 is called square-free.

Square root

y is a square root of number x if y2= x

Every positive number x = y2has two square roots, +y and -y.

Square roots of positive whole numbers that are not perfect squares are always irrational numbers

Perfect numbers

If the sum of all the positive factors of a number (known as aliquot sum) is equal to the number itself, then the number is a perfect number.

Example: 6 (factors are 1, 2, 3 and 6). 1+2+3 = 6
28 is a perfect number as 1 + 2 + 4 + 7 + 14 = 28

It is unknown whether there are any odd perfect numbers.

A Square number cannot be a perfect number.

The only even perfect number of the form x>3 + 1 is 28. 28 is also the only even perfect number that is a sum of two positive integral cubes (27 + 1).

The reciprocals of the divisors of a perfect number N must add up to 2
(for 6, 1/6 + 1/3 + 1/2 + 1 = 2).

Euclid proved that 2p1(2p−1) is an even perfect number whenever 2p−1 is prime

for p = 2:   21(22−1) = 6
for p = 3:   22(23−1) = 28
for p = 5:   24(25−1) = 496
for p = 7:   26(27−1) = 8128.

Co Primes (or relatively primes)

Two numbers are said to be co primes if the only common factor between then is one.
E.g.: (5, 4), (3, 10)

Two consecutive integers are always co primes.

Complex numbers

a number which can be put in the form of a + i b where a and b are real numbers and i is called the imaginary unit

i= √-1. ( i2= -1 )

Triangular numbers

Sum of n consecutive integers starting with 1.
E.g.: 1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10

Why it is called Triangular numbers? Because using ‘triangular number’ of objects we can form an equilateral triangle. Check it out!

All even perfect numbers (till we get our odd one) is a triangular number.

Armstrong number(narcissistic number)

Is an integer such that the sum of their own digits to the power of the number of digits is equal to the number itself. 371 = 3³+ 7³+ 1³, 153 = 1³+ 5³+ 3³

Kaprekar number

Consider n digit number K. Square the number and if the sum first n digits from rightand the n ( or n-1) digits from left of K2 is equal to K it is a Kaprekar number.

For example: K = 9, n =1. K2 = 81, now 8 + 1 = 9 = K;
K = 703, n =3. K2= 494209, 494 + 209 = 703 = K.

Bell number

The number of ways a set of n elements can be partitioned into nonempty subsets is called a Bell number.
For example, there are five ways the numbers {1,2,3} can be partitioned into nonempty sets: {{1},{2},{3}}, {{1,2},{3}}, {{1,3},{2}}, {{1},{2,3}}, and {{1,2,3}}, so Bellnumber(3) = 5.

Palindromes

A palindrome number is a number that is the same when written forwards or backwards. Example: 44, 121, 8778

Automorphic numbers

A number k such that n.k2 has its last digit(s) equal to k is called n-automorphic.

For example, 1·52=25 and 1·62=36, so 5 and 6 are 1-automorphic.

82=128 and 2·882=15488, so 8 and 88 are 2-automorphic.

Ramanujan number ( 1729 )

It is the smallest number expressible as the sum of two positive cubes in two different ways.

1729 = 13+ 123= 93+ 103

Smith number

A Smith number is a composite number the sum of whose digits is the sum of the digits of its prime factors.

The primes are excluded since they trivially satisfy this condition.

Example of a Smith number is the Beast number (666)

666=2·3·3·37

6+6+6=2+3+3+ (3+7) =18.

Amicable Pair

An amicable pair (m, n) consists of two integers m, n for which the sum of proper divisors (the divisors excluding the number itself) of one number equals the other.

Example: (220,284)

Friedman number

A Friedman number is a positive integer which can be written in some non-trivial way using its own digits, together with the symbols + - x / ^ ( ) and concatenation. For example, 25 = 52

Sharing some Friedman numbers, can you find out how the digits of below numbers can be used to generate the number itself with the operations allowed.

121, 125, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024

Some useful properties of numbers

Sum of first n natural numbers = n * (n + 1) / 2

Sum of first n odd natural numbers is equal to n2

Sum of first n even natural numbers is equal to n (n + 1)

Sum of squares of first n natural numbers = n (n + 1) (2n + 1) / 6

Sum of cubes of the first n natural nos. = [n (n + 1) / 2] 2

even ± even = even

even ± odd = odd

odd ± odd = even

even × even = even

even × odd = even

odd × odd = odd

Goldbach's conjecture Every even integer greater than 2 can be represented as a sum of two prime numbers ; 12 = 5 + 7

Any positive integer can be written as the sum of four or fewer perfect squares.
14 = 4 + 1 + 9 (Lagrange's four-square theorem)

Any prime number of  the form (4n + 1) can be written as sum of two squares.
17 = 16 + 1, 41 = 25 + 16

A perfect square can end only with digit 1, 4, 6, 9, 5 or even number of zeros.

Adding the digits of perfect square will always yield 1, 9, 4 or 7. A number is not a perfect square if its digit sum does not evaluate to one of these. Number may or may not be a perfect square if its digit sum is 1, 9, 4 or 7.
Digit sum (1936) = digit sum (1 + 9 + 3 + 6 = 19) = digit sum (1 + 9 = 10) = 1

All prime numbers can be represented as (6n-1) or (6n+1). So if we cannot represent a number in (6n+1) or (6n-1) form we can say it is not a prime. The reverse may not be true, i.e. all numbers that can be represented as (6n-1) or (6n+1) need not be a prime.

Any odd number is a difference of two consecutive squares
(k + 1)2= K2+ 2k +12, so (k + 1)2- k2= 2k + 1 (an odd number)

Any multiple of 4 is a difference of the squares of two numbers that differ by two
(k + 2)2– k2= 4k + 4 = 4 (k +1)

30, 31… 3n can produce all integers from 1 to 1 + 31 + ... + 3n

Let’s play …

1. Three times the first of three consecutive odd numbers is 3 more than twice the third. What is the third integer?

2. If 8 + 12 = 2, 7 + 14 = 3 then 10 + 18 =?

3. If 38 pieces of clothing (including shirts, trousers and ties) are sold where at least 11 items in each category is sold. What will be the number of ties must have sold if more shirts are sold than trousers and more trousers than ties?

4. A 2 digit number exceeds by 19 the sum of the squares of the digits and by 44 the double product of its digits. Find the number.

5. 95, 52, 43, 148, 62, 86, 196, __, 121.

6. 24:15:: 63: __

7. 0, 6, 24, 60, 120, ___

8. 6, 14, 36, 98, 276, ___

9.  5 + 3 + 2 = 151022
9 + 2 + 4 = 183652
8 + 6 + 3 = 482466
5 + 4 + 5 = 202541
7 + 2 + 5 =?

10. Try making 1000 with eight 8s and only addition is allowed.

11. Create an eight digit number using the digits 4, 4, 3, 3, 2, 2, 1 and 1. So that, 1s are separated by 1 digit, 2s are separated by 2 digits, 3s are separated by 3 digits and 4s are separated by 4 digits.

12. It takes 1629 digits to number the pages of a book. How many pages does the book have?

13. What is the minimum number of weights required to weigh from 1kg to 40 kg (integer values only)

14. Prove that 121 is a Friedman number (Obtain the number 121 from the digits of the number (1, 2 and 1) and mathematical operations. Each digit can be used only once)

15. Use exactly four 4's to form every integer from 0 to 10, using only the operators +, -, *, /, () (brackets) x2 (square) Factorial (4! = 24) and decimal point.

Solutions

1. 15.
Take the numbers as 2n+1, 2n+3 and 2n+5
3* (2n+1) = 2 * (2n+5) + 3
Solve and we get n = 5, so third integer is 2*5 + 5 = 15.

2. Sometimes we may not see the direct answer in the options. Answer will be again manipulated through some method. Here what we need is the sum of the digits in the answer, 2 + 8 = 10

3. 11 ties are sold.
Consider x Shirts, y trousers and z ties are sold
x + y + z = 38; x, y, z >=11; x > y > z
only possible answer is x = 14, y = 13 and z = 11.

4. 72.
Let the digits be a and b, number can be written as 10a + b
10a + b = a2+ b2 + 19 --- (1)
10a + b = 2ab + 44 --- (2)
(1)  – (2), a - b = 5, a = b+5, replace ‘a’ in (1), we get b=2, a=2+5=7

5. 75. Series can be obtained as 95 - 52 = 43, 148 - 62 = 86,196 - 75 = 121

6. 24:15 = 52-1: 42-1; 63 = 82-1, hence the missing number is 72-1 = 48

7. 210.  Series can be obtained as nth term = (n-1) n (n+1)

8. 794. Series can be obtained as nth term = 1n + 2n + 3n

9. 143547 (7 * 2, 7 * 5, 7 * 2 + 7 * 5 -2)

10. 888 + 88 + 8 + 8 + 8 = 1000

11. 41312432; 23421314

12. Page1 to Page9 we need 9 digits. Page10 to Page99 we need 180 digits. We are left with 1629 – 180 – 8 = 1440 which can gives us 480 (1440/3) pages with 3 digit page numbers (100-579), we have total 579 pages.

13.  4 weights, 1 kg, 3 kg, 9 kg, 27kg. We can measure all values from 1 and 40. e.g.: to measure 5 kg have 9kg weight at one pan and 3 kg + 1 kg weight on the other.

14. 121 = 112

15.   0 = 44 – 44; 1 = 44 / 44 ; 2 = 4/4 + 4/4 ; 3 = (4 + 4 + 4) / 4; 4 = 4 * (4 - 4) + 4 ; 5 = (4 * 4 + 4) / 4 ; 6 = 4 * .4 + 4.4 ; 7 = (44 / 4) – 4 ; 8 = 4 + 4.4 - .4 ; 9 = 4/4 + 4 + 4 ; 10 = 44 / 4.4

I am sure you know more number types and number properties.. do share :-)

Happy Learning! :)

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