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Math Lesson on Algebra & Number Theory

BASIC PROPERTIES OF NUMBERS • Absolute value (of a real number): The number’s distance from zero (the origin) on the real-

number line. The absolute value of x is indicated as |x|. (The absolute value of a negative

number can be less than, equal to, or greater than a positive number.)

• Integer: Any non-fraction number on the number line: {. . . 23, 22, 21, 0, 1, 2, 3 . . .}. Except

for the number zero (0), every integer is either positive or negative. Every integer is either even

or odd.

• Factor (of an integer n): Any integer that you can multiply by another integer for a product of

n.

• Prime number: Any positive integer greater than one that has exactly two positive factors: 1

and the number itself. In other words, a prime number divisible by (a multiple of) any positive

integer other than itself and 1.

Number Signs and the Four Basic Operations

N.B: “?” indicates that the sign depends on which number has the greater absolute value): Example: 01

A number M is the product of seven negative numbers, and the number N is the product of six negative numbers and one positive number. Which of the following holds true for all possible values of M and N? I. M X N < 0 II. M - N < 0 III. N + M < 0 (A) I only (B) II only (C) I and II only (D) II and III only

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(E) I, II, and III Solution: The product of seven negative numbers is always a negative number. (M is a negative number.)

The product of six negative numbers is always a positive number, and the product of two

positive numbers is always a positive number. (N is a positive number.) Thus, the product of M

and N must be a negative number; I is always true. Subtracting a positive number N from a

negative number M always results in a negative number less than M; II is always true. However,

whether III is true depends on the values of M and N. If |N|> |M|, then N + M > 0, but if |N|<

|M|, then N+ M < 0

The correct answer is (C)

Integers and the Four Basic Operations ADDITION AND SUBTRACTION • Integer ± integer = integer • Even integer ± even integer = even integer • Even integer ± odd integer = odd integer • Odd integer ± odd integer = even integer MULTIPLICATION AND DIVISION • Integer X integer = integer • Integer ÷ non-zero integer = integer, but only if the numerator is divisible by the denominator (if the result is a quotient with no remainder) • Odd integer X odd integer = odd integer • Even integer X non-zero integer = even integer • Even integer ÷ 2 = integer • Odd integer ÷ 2 = non-integer

Example: 02

If P is an odd integer and if Q is an even integer, which of the following expressions CANNOT

represent an even integer?

(A) 3P - Q (B) 3P X Q (C) 2Q X P (D) 3Q - 2P (E) 32P - 2Q Solution: Since 3 and P are both odd integers, their product (3P) must also be an odd integer. Subtracting an even integer (Q) from an odd integer results in an odd integer in all cases

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The correct answer is (A).

FACTORS, MULTIPLES, AND DIVISIBILITY Figuring out whether one number (f) is a factor of another (n) is no big deal. Just divide n by f. If

the quotient is an integer, then f is a factor of n (and n is divisible by f). If the quotient is not an

integer, then f is not a factor of n, and you’ll end up with a remainder after dividing. For

example, 2 is a factor of 8 because 8 ÷2 = 4, which is an integer. On the other hand, 3 is not a

factor of 8 because 8 ÷ 3 = 8/3 w

Which is a non-integer. (The remainder is 2.)

Remember these four basic rules about factors, which are based on the definition of the term

“factor”:

Any integer is a factor of itself.

1 and -1 are factors of all integers.

The integer zero has an infinite number of factors but is not a factor of any integer.

A positive integer’s greatest factor (other than itself) will never be greater than one half

the value of the integer.

On the “flip side” of factors are multiples. If f is a factor of n, then n is a multiple of f. For

example, 8 is a multiple of 2 for the same reason that 2 is a factor of 8—because 8 ÷2 = 4, which

is an integer.

Example: 03

If n > 6, and if n is a multiple of 6, which of the following is always a factor of n?

(A) n - 6

(B) n + 6

(C) n / 3

(D) n /2 + 3

(E) n/2 + 6

Solution:

Since 3 is a factor of 6, 3 is also a factor of any positive-number multiple of 6. Thus, if you divide

any multiple of 6 by 3, the quotient will be an integer. In other words, 3 will be a factor of that

number (n).

As for the incorrect choices, n - 6 (choice (A)) is a factor of n only if n = 12. n + 6 (choice (B)) can

never be a factor of n because n + 6 is greater than n. You can eliminate choices (D) and (E)

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because the greatest factor of any positive number (other than the number itself) is half the

number, which in this case is n/2

Another strategy works for the preceding question; you should tryout more than one sample

value for n. If n 5 12, choices (A), (C), and (E) are all viable. But try out the number 18, and

choice (C) is the only factor of n. (To be on the safe side, you should try out at least one

additional sample value as well, such as 24.)

The correct answer is (C).

Divisibility Rule:

A number is divisible by

Check to see

2 If it is even

3 If the sum of the digits is divisible by 3

4 If the number formed by last two digits is divisible by 4

5 If it ends by 5 or 0

6 If it is even and divisible by 3

7 Form the alternating sum of blocks of three from right to left as per below example: 1,369,851: 851-369 +1 = 483.

If the resulting number (483 in this case) is divisible by 7, then the entire number will be divisible by 7

8 If the number formed by last three digits is divisible by 8

9 If the sum of the digits is divisible by 9

10 If it ends by 0

11 Alternately add and subtract the entire digits of the given number and if the result is a multiple of 11, the entire number is divisible by 11.

Example: 1,369,851 = +1-3+6-9+8-5+1 = -1 ; however -1 is not a multiple of 11, so the number 1,369,851 can’t be divided by 11

12 It has to be divisible by 3 & 4 both

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13 Same like the rule of 7’s divisibility

14 It is even and divisible by 7

15 It is divisible by 3 & 5

16 If the number formed by last four digits is divisible by 16

17 Subtract 5 times the last digit from the rest. If the resulting

number is divisible by 17, the entire number is divisible by

17

Example : 221 : 22- 1 X 5 = 17

18 It is even and divisible by 9

19 Add twice the last digit to the rest. If the resulting number

is divisible by 19, the entire number is divisible by 19

Example : 437 : 43+ 7 X 2 = 57

20 If the number formed by the last two digits is divisible by

20

PRIME NUMBERS AND PRIME FACTORIZATION A prime number is a positive integer greater than one that is divisible by only two positive

integers: itself and 1. Just for the record, here are all 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

Remember:

Only 2 is the even prime number.

Neither 0 nor 1 is considered prime number

Prime number testing:

If you want to test any number more than 100, whether it is prime or not, take an integer larger

than the approximate square root of that number. Let it be x. Test the divisibility of the given

number by every prime number less than x. If it is not divisible by any of them, then it is prime,

otherwise it is a composite number.

To find what’s called the prime factorization of a non-prime integer, divide the number by the

primes in order and use each repeatedly until it is no longer a factor. For example:

110 = 2 X 55

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= 2 X 5 X 11. This is the prime factorization of 110.

Stop when all factors are prime and then if a factor occurs more than once, use an exponent to

indicate this (i.e., write it in exponential form.)

EXPONENTS (POWERS) An exponent, or power, refers to the number of times a number (referred to as the base) is

used as a factor. In the number 23, the base is 2 and the exponent is 3. To calculate the value of

23, you use 2 as a factor three times: 23= 2 X 2 X 2 = 8.

Combining Exponents by Addition or Subtraction When you add or subtract terms, you cannot combine bases or exponents. It’s as simple as

that.

Ax + Bx ≠ (A + B)x

Ax- Bx ≠ (A - B)x

Combining Exponents by Multiplication or Division It’s a whole different story for multiplication and division. First, remember these two simple

rules:

You can combine bases first, but only if the exponents are the same:

Ax X Bx = (AB) x

You can combine exponents first, but only if the bases are the same. When multiplying

these terms, add the exponents. When dividing them, subtract the denominator

exponent from the numerator exponent:

Ax X Ay = (A)x+y

Ax / Ay = (A)x-y

When the same base appears in both the numerator and denominator of a fraction, you

can

Cancel the number of powers common to both.

Additional Rules for Exponents To cover all your bases, also keep in mind these three additional rules for exponents:

When raising an exponential number to a power, multiply exponents:

(Ax)y = Axy

Any number other than zero (0) raised to the power of zero (0) equals 1:

A0 = 1 [A ≠ 0]

Raising a base other than zero to a negative exponent is equivalent to 1 divided by the

base raised to the exponent’s absolute value:

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A-x = 1/Ax

Exponents You Should Know

Number Line: In basic mathematics, a number line is a picture of a straight line on which every

point is assumed to correspond to a real number and every real number to a point. Often the

integers are shown as specially-marked points evenly spaced on the line. Although this image

only shows

the integers from −9 to 9, the line includes all real numbers, continuing forever in each

direction, and also numbers not marked that are between the integers. It is divided into two

symmetric halves by the origin, i.e. the number zero.

Fractions Fractions consist of two numbers separated by a line called a fraction bar. The number above the line is

called the numerator and the number below the line is called the denominator. In the fraction 4/5 , 4 is

the numerator and 5 is the denominator. If numerator < denominator, the fraction is called proper

function. If numerator > denominator, the fraction is known as improper function.

A mixed number is a number composed of a whole number and a proper fraction. It is always greater than 1 in value: 7 3 ----- 8

Remember, when a positive fraction less than 1 is squared, the result is smaller than the original

number. Again, when a square root of a positive fraction is done, it is always greater than the original

number!

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Least Common Multiple (abbreviated L.C.M.) of two natural numbers is the smallest natural

number which is a multiple of both the numbers.

LCM of Fractions:

LCM of fractions = LCM of numerators/HCF of denominators.

Highest Common Factor (abbreviated H.C.F.) of two natural numbers is the largest common

factor (or divisor) of the given natural numbers. In other words, H.C.F. is the greatest element

of the set of common factors of the given numbers. H.C.F. is also called Greatest Common

Divisor (abbreviated G.C.D.)

HCF of fractions:

HCF of fractions = HCF of numerators/LCM of denominator

Co-prime numbers: Two natural numbers are called co-prime numbers if they have no

common factor other than 1.In other words, two natural numbers are co-prime if their H.C.F. is

1. Some examples of co-prime numbers are: 4, 9; 8, 21; 27, 50.

Relation between L.C.M. and H.C.F. of two natural numbers:

The product of L.C.M. and H.C.F. of two natural numbers = the product of the numbers.

Note. In particular, if two natural numbers are co-prime then their L.C.M. = the product of the

numbers.

Type of questions on LCM & HCF and corresponding approach to solve those:

Type of Question Approach

Type 1 : Find the least number, which is exactly divisible by x, y,

z.

LCM (x, y, z)

Type 2: Find the least number, which when divided by x, y, z,

leaves a remainder ‘r’ in each case.

LCM (x, y, z) + r

Type 3: Find the least number, which when divided by x, y, z

leaves remainders a, b, c respectively.

Observe, if x – a = y – b = z –

c = k (say). Then LCM (x, y,

z) – k. Else use options.

Type 4: Find the greatest number, that will exactly divide x,y,z. HCF (x, y, z)

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Type 5: Find the greatest number, that will divide x, y, z leaving

remainders a, b, c respectively.

HCF (x – a, y – b, z – c)

Type 6: Find the greatest number, that will divide x, y, z leaving

the same remainder in each case.

HCF (x – y, y – z, z – x)

Now, based on the table above, try to solve the below mentioned HCF & GCF problems:

1. Find the largest number which can divide 284, 698 & 1618 leaving the same remainder 8

in each case. [answer is 46]

2. Find the largest number which divides 62, 132 and 237 to leave the same remainder in

each case. [answer is 35]

3. Find the largest number of four digits divisible by 12, 15, 18 and 27 [answer is 9720]

4. Find the smallest number of five digits exactly divisible by 16, 24, 36 and 54 [answer is

10368]

5. The greatest number by which if 1657 and 2037 are divided to give remainders 6 and 5

respectively is what? [answer is 127]

6. The smallest number which when increased by 5 is divisible by each one of 24, 32, 36 &

54 is what? [answer is 859]

7. The least number which when divided by 5,6,8,9 and 12 leaves a remainder 1 each case

but when divided by 13, leaves no remainder. Then the number is what?

Solution: LCM of 5,6,8,9 and 12 = 360

So, required number will be in the form of = 360K + 1

From observation, least value of K for which 360K+1 is divisible by 13 is K=10.

So, the number is = 360 x10 + 1 = 3601

8. Six bells commence tolling together and toll at intervals of 2, 4,6,8,10 and 12 seconds

respectively. In 30 minutes, how many times do they toll together?

Solution: LCM of 2, 4,6,8,10 and 12 = 120

So, the bells will toll together after every 120 seconds or 2 minutes. So, after tolling

together for the first time, they will toll together total = 30/2 =15 times

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So, total tolling = 15+1 = 16 times (answer)

Reciprocals

The reciprocal of a fraction is another fraction with the numerator and denominator reversed.

The reciprocal of , for instance, is . The product of a fraction and its reciprocal is 1; hence

the reciprocal is the multiplicative inverse of a fraction.

Problems for practice in the class

1. If the remainder is 1 when m is divided by 2 and the remainder is 3 when n is divided by 4,

which of the following must be true?

(A) m is even. (B) n is even. (C) m + n is even. (D) mn is even. (E) m/n is even.

2. If x and y are both prime and greater than 2, then which of the following CANNOT be a

divisor of xy?

(A) 2 (B) 3 (C) 11 (D) 15 (E) 17

3. If 2 is the greatest number that will divide evenly into both x and y, what is the greatest

number that

Will divide evenly into both 5x and 5y?

(A) 2 (B) 4 (C) 6 (D) 8 (E) 10

4. If x is both the cube and the square of an integer and x is between 2 and 200, what is the

value of x?

(A) 8 (B) 16 (C) 64 (D) 125 (E) 169

5. In the two-digit number x, both the sum and the difference of its digits is 4. What is the value

of x?

(A) 13 (B) 31 (C) 40 (D) 48 (E) 59

6. Which one of the following numbers is the greatest positive integer x such that 3x is a factor

of 275?

(A) 5 (B) 8 (C) 10 (D) 15 (E) 19

7. If the sum of two prime numbers x and y is odd, then the product of x and y must be divisible

by

(A) 2 (B) 3 (C) 4 (D) 5 (E) 8

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8. If (x + y) /(x – y) = 3 and x and y are integers, then which one of the following must be true?

(A) x is divisible by 4

(B) y is an odd number

(C) y is an even integer

(D) x is an even number

(E) x is an irreducible fraction

9. Which one of the following could be the difference between two numbers both of which are

divisible

By 2, 3 and 4?

(A) 71 (B) 72 (C) 73 (D) 74 (E) 75

10. Which of the following is the largest? [For this problem, course instructor will show

students---the gap rule, cross rule of factor comparison, process of making denominator same

for fractions and rule of deciding by observation—i.e total 4 methods]

[EMBA April 2010-18th Batch]

(A) 1/2 (B) 7/15 (C) 126/250 (D) 1999/4000 (E) All are equal

11. If the value of X & Y in the fraction XZ/Y is both tripled, how does the value of the fraction

change? [IBA MBA 2009-2010]

(A) Increases by half (B) Decreases by half (C) Triples (D) Doubles (E) Remains the same

12. When positive integer m is divided by positive integer n, the remainder is 6. If m/n = 112.15,

what is the value on n? [South East Bank—2009, DBBL—2009]

(A) 15 (B) 20 (C) 40 (D) 25 (E) None of these

13. When 20 is divided by the positive integer k, the remainder is (k-2). Which of the following

is a possible value of k? [Basic Bank—2009]

(A) 8 (B) 9 (C) 10 (D) 11 (E) None of these

Answer Keys

1. C 2. A 3. E 4. C 5. C 6. D 7. A 8. D 9. B 10. C 11. E 12. C 13. D