Guys this group i created for discussion related to all topics. Hope we all participate actively and discuss in depth each and every question , nd finally the winners of cat 2013. Help out each other in queries is main motive of this thread :smil…

Guys this group i created for discussion related to all topics. Hope we all participate actively and discuss in depth each and every question , nd finally the winners of cat 2013. Help out each other in queries is main motive of this thread 😃 😃 ...ATB to all and request to join 😃

Remainder theorem : important concept on that

When the product of any two or more natural numbers is divided by any natural number then it leaves the same remainder as the product of the individual remainders i.e. if a, b , c and d are integers and k is a positive integer such that,

a = c (mod k)

And, b = d (mod k)

Then, a*b = c*d (mod k)

The same holds true for the other operations as well such as addition, subtraction and division.

i.e. a+b = c+d (mod k)

a-b = c-d (mod k)

(a/b) = (c/d) (mod k)

For example, 75 (= 15*5) which when divided by 4 leaves the remainder of -1 (or 3). And if we consider the product of individual remainders for 15 and 5 with 4, i.e. (-1)*1 which is same when 75 is directly divided by 4.

hope it will be useful

most important thing regarding abt result is that , we are not bounded by any condition that either of a,b,c,d will be any prime no or anything ...only thing is that they should be integer 😃

what are relatively prime/ coprime numbers?

A set of numbers which do not have any other common factor other than 1 are called co-prime or relatively

prime numbers. This means those numbers whose HCF is 1.

For example, 8 and 9 have no other common factor other than 1 so they are co-prime numbers.

What is a coprime numberProperties Of Co-prime Numbers:• All

prime numbersare co-prime to each other.

• Any 2 consecutive

integersare always co-prime.

• Sum of any two co-prime numbers is always co-prime with their product.

• 1 is co-prime with all numbers.

• a and b (

natural numbers) are co-prime only if the numbers 2a-1 and 2b-1 are co-prime.

For any number n (= p^a*q^b*r^c*_____, where p, q, r,____ are prime numbers), Euler's Totient Function is determined as, Φ(n) = n*(1 – 1/p)*(1 – 1/q)*(1 – 1/r)*____

For example, for n = 12 (= 2^2*3), Φ(n) will be 12*(1 – 1/2)*(1 – 1/3) i.e. 4 which means 12 has 4 numbers (1, 5, 7 and 11) less than itself which are co-prime to it.

**Euler's Theorem**- If two numbers N and K are co-prime to each other,then K^Φ(n) = 1modN

where Φ(n) is the Euler's Totient Function of N.

For example 1), 37^4 when divided by 12, will leave a remainder of 1 as

**Euler's Totient Function**of 12 is 4.

2) In order to find the remainder when 41^97 is divided by 12,

41^97 = 41*41^96

Now as 41^96 = (41^4)^24 = 1mod12 (as 41^4 = 1mod12)

And 41 = 5mod12

=> From the

**basic remainder principle**, 41^97 = (41mod12)*(41^96mod12) = 5*1mod12 = 5mod12.

**Fermat's Theorem**- For a prime number P, the

Euler's Totient Functionwill be P(1- 1/P) i.e.equal to (P-1).

So from

**Euler's theorem**, K^(P €“ 1) = 1modP.

For example 1), 33^12 will leave a remainder of 1 when divided by 13.

2) In order to find the remainder when 43^37 is divided by 13,

43^37 = 43*43^36

Now 43^36 = (33^12)^3 = 1mod13

And 43 = 4mod13.

So, from

**basic remainder principle**, 43^37 = 4*1mod13 = 4mod13.

to calculate remainder for factorial type questions :**Wilson's Theorem**-(p €“ 1)! + 1 is divisible by p if p is a prime number. We can say this also that the remainder when (p-1)! is divided by p, will be -1 or (p-1).

Remember, If p is an integer greater than one then p is prime if and only if **(p-1)! = -1 (mod p).**

Example- What will be the remainder when 28! is divided by 29?

Solution- As 29 is a prime number. So from **Wilson's theorem** we can say that 28!+1 will be divisible by 29 or 28! will leave a remainder of -1 (i.e. 28)when divided by 29.**Rem (p-2)!/p = 1, where p is a prime number.**

for polynomial type questions on remainder theorem :**Remainder Theory for Polynomials**- Say, q(x) and r(x) are the quotient and remainder, respectively, when the polynomial f (x) (= a + bx + cx^2 + dx^3 +..) is divided by x − a,

then f (x) = (x − a)q(x) + r(x).

The degree of the remainder will be less than that of the divisor, hence remainder must be constant

So, f (x) = (x − a)q(x) + k.

Substituting x = a in the above equation,

f(a) = k.

Hence, when a polynomial f(x) is divided by (x – a), then the remainder is equal to f (a).

For example

1) When 5x^3 + 7x – 9 is divided by x – 2 will leave a remainder of 5*(2)^3 + 7*2 – 9 i.e. equal to 45.

2) What is the remainder when (81)^21 + (27)^21 + (9)^21 + (3)^21 + 1 is divided by 3^20 + 1?

Solution) Let 3^20 = x

=> We have to find the remainder when f(x) = 81x^4 + 27x^3 + 9x^2 + 3^x + 1 is divided by x + 1.

=> f(-1) = 81 - 27 + 9 - 3 + 1 = 61.

3) What will be the remainder when f(x) = x^71 + x^50 + x^25 + x^9 is divided by x^3 − x ?

Solution) As the degree of the divisor is 3, degree of the remainder will be less than or equal to 2.

=> Say, the remainder is ax^2 + bx + c.

Now, x^71 + x^50 + x^25 + x^9 = q(x)*(x^3 – x) + ax^2 + bx + c = x*(x – 1)*(x + 1)*q(x) + (ax^2 + bx + c).

=> f(0) = 0 = c

f(1) = 4 = a + b + c => a + b = 4

And f(-1) = -1 + 1 -1 -1 = -2 => a – b = -2

=> a = 1 and b = 3

Hence the remainder will be (x^2 + 3x).

**CAT 2012**: 104^303 is divided by 101. remainder???

solution : first we look that the divisor is prime number , so we can easily move to fermat theorem

104^303 = (104^3) * (104^300) --(1)

104^300 = (104^100)^3--(2)

for 104^100 we can apply fermat theorem

[a^(p-1)] = 1 mod p for p is prime number

104^100 = 1 mod 101

put this in eqn 2

104^300 = 1 mod 101

now 104 = 3mod 101

so putting these two values in eqn 1

we get

104^303 = (3 mod 101)^3 * 1

= 27 ANSWER :)

9^1 +9^2 +..............+9^8

is divided by 6?

3

2

0

5

Usage of however & nevertheless (adverbs) : used to express contrasting points**However and nevertheless: to express a contrast**

We can use either of the adverbs **however** or **nevertheless** to indicate that the second point we wish to make contrasts with the first point. The difference is one of formality: **nevertheless** is bit **more formal** and emphatic than however. Consider the following:*I can understand everything you say about wanting to share a flat with Martha. However, I am totally against it. **Rufus had been living in the village of Edmonton for over a decade. Nevertheless, the villagers still considered him to be an outsider.*

@rmaheshwari33 said:What is the remainder when9^1 +9^2 +..............+9^8is divided by 6?3205

each leaves a remainder of 3

3^n = 3 mod 6

so 3*8=24

which is div by 6

so 0

@rudra13 : thanks ...

y=1/3 +3/18+15/162+........... find the value of y^2+2y.

a)

**2**

b) 1

c) 1.5

d) None

right ans is 2

a) (8!)^8!+(9!)^9!+(10!)^10!+(11!)^11!

b)

**(10)^101**

c) 4!+6!+8!+2(10!)

d) (0!)^0!

right answer is d

Factorials & trailing zeroes:

we get 0 when there is multiple of 10 (5 *2) , and we get extra 5's when there is 25 (5 *5 )

so **Find the number of trailing zeroes in ****101!**

Okay, how many multiples of 5 are there in the numbers from 1 to 101? There's 5, 10, 15, 20, 25,...

Oh, heck; let's do this the short way: 100 is the closest multiple of 5 below 101, and 100 ÷ 5 = 20, so there are twenty multiples of 5 between 1 and 101.

But wait: 25 is 5×5, so each multiple of 25 has an extra factor of 5 that I need to account for. How many multiples of 25 are between 1 and 101? Since 100 ÷ 25 = 4, there are four multiples of 25 between 1 and 101.

Adding these, I get 20 + 4 = **24**** trailing zeroes in ****101!**

This reasoning extends to working with larger numbers.**Find the number of trailing zeroes in the expansion of ****1000!**

Okay, there are 1000 ÷ 5 = 200 multiples of 5 between 1 and 1000. The next power of 5, namely 25, has 1000 ÷ 25 = 40 multiples between 1 and 1000. The next power of 5, namely 125, will also fit in the expansion, and has 1000 ÷ 125 = 8 multiples between 1 and 1000. The next power of 5, namely 625, also fits in the expansion, and has 1000 ÷ 625 = 1.6... um, okay, 625 has 1 multiple between 1 and 1000. (I don't care about the 0.6 "multiples", only the one full multiple, so I truncate my division down to a whole number.)

In total, I have 200 + 40 + 8 + 1 = 249 copies of the factor 5 in the expansion, and thus:**249**** trailing zeroes in the expansion of ****1000!**

The previous example highlights the general method for answering this question, no matter what factorial they give you.

Take the number that you've been given the factorial of.

Divide by 5; if you get a decimal, truncate to a whole number.

Divide by 52 = 25; if you get a decimal, truncate to a whole number.

Divide by 53 = 125; if you get a decimal, truncate to a whole number.

Continue with ever-higher powers of 5, until your division results in a number less than 1. Once the division is less than 1, stop.

Sum all the whole numbers you got in your divisions. This is the number of trailing zeroes.

Here's how the process works:**How many trailing zeroes would be found in ****4617!****, upon expansion?**

I'll apply the procedure from above:

51 : 4617 ÷ 5 = 923.4, so I get 923 factors of 5

52 : 4617 ÷ 25 = 184.68, so I get 184 additional factors of 5

53 : 4617 ÷ 125 = 36.936, so I get 36 additional factors of 5

54 : 4617 ÷ 625 = 7.3872, so I get 7 additional factors of 5

55 : 4617 ÷ 3125 = 1.47744, so I get 1 more factor of 5

56 : 4617 ÷ 15625 = 0.295488, which is less than 1, so I stop here.

Then **4617!**** has ****923 + 184 + 36 + 7 + 1 = 1151**** trailing zeroes.**

some intresting points for subject verb agreement :

### Basic Rule

The basic rule states that a singular subject takes a singular verb, while a plural subject takes a plural verb.

**NOTE:** The trick is in knowing whether the subject is singular or plural. The next trick is recognizing a singular or plural verb.

** Hint**: Verbs do not form their plurals by adding an s as nouns do. In order to determine which verb is singular and which one is plural, think of which verb you would use with

*he*or

*she*and which verb you would use with

*they*.

**Example:**

talks, talk

Which one is the singular form?

Which word would you use with *he*?

We say, "He talks." Therefore, *talks* is singular.

We say, "They talk." Therefore, *talk* is plural.

### Rule 1

Two singular subjects connected by *or* or *nor* require a singular verb.

**Example:***My aunt or my uncle is arriving by train today.*

### Rule 2

Two singular subjects connected by *either/or* or *neither/nor* require a singular verb as in Rule 1.

**Examples:***Neither Juan nor Carmen is available.*

*Either*

__Kiana__or__Casey____is helping__today with stage decorations.### Rule 3

When* I* is one of the two subjects connected by *either/or* or *neither/nor*, put it second and follow it with the singular verb *am*.

**Example:***Neither she nor I am going to the festival.*

### Rule 4

When a singular subject is connected by *or* or *nor* to a plural subject, put the plural subject last and use a plural verb.

**Example:***The serving bowl or the plates go on that shelf.*

### Rule 5

When a singular and plural subject are connected by *either/or* or *neither/nor*, put the plural subject last and use a plural verb.

**Example:***Neither Jenny nor the others are available.*

### Rule 6

As a general rule, use a plural verb with two or more subjects when they are connected by *and*.

**Example:**

A __car__ and a __bike__ __are__ my means of transportation.

### Rule 7

Sometimes the subject is separated from the verb by words such as *along with, as well as, besides*, or *not*. Ignore these expressions when determining whether to use a singular or plural verb.

**Examples:***The politician, along with the newsmen, is expected shortly.*

Excitement, as well as nervousness,

__is__the cause of her shaking.

### Rule 8

The pronouns *each, everyone, every one, everybody, anyone, anybody, someone,* and *somebody* are singular and require singular verbs. Do not be misled by what follows *of*.

**Examples:**__Each__ of the girls __sings__ well.*Every one of the cakes is gone.*

**NOTE:** *Everyone* is one word when it means *everybody*. *Every one* is two words when the meaning is *each one*.

### Rule 9

With words that indicate portions €”*percent, fraction, part, majority, some, all, none, remainder*, and so forth €”look at the noun in your *of* phrase (object of the preposition) to determine whether to use a singular or plural verb. If the object of the preposition is singular, use a singular verb. If the object of the preposition is plural, use a plural verb.

**Examples:***Fifty *__percent__ of the pie __has__ disappeared.*Pie* is the object of the preposition *of*.*Fifty *__percent__ of the pies __have__ disappeared.*Pies* is the object of the preposition.__One-third__ of the city __is__ unemployed.__One-third__ of the people __are__ unemployed.

**NOTE:** Hyphenate all spelled-out fractions.

__All__ of the pie __is__ gone.__All__ of the pies __are__ gone.__Some__ of the pie __is__ missing.__Some__ of the pies __are__ missing.*None** of the garbage was picked up.*

*None**of the sentences*

__were punctuated__correctly.*Of all her books,*

__none____have sold__as well as the first one.**NOTE:** Apparently, the SAT testing service considers *none *as a singular word only. However, according to *Merriam Webster's Dictionary of English Usage,* "Clearly *none* has been both singular and plural since Old English and still is. The notion that it is singular only is a myth of unknown origin that appears to have arisen in the 19th century. If in context it seems like a singular to you, use a singular verb; if it seems like a plural, use a plural verb. Both are acceptable beyond serious criticism" (p. 664).

### Rule 10

The expression *the number *is followed by a singular verb while the expression *a number* is followed by a plural verb.

**Examples:**__The number__ of people we need to hire __is__ thirteen.__A number__ of people __have__ written in about this subject.

### Rule 11

When *either* and *neither* are subjects, they always take singular verbs.

**Examples:**__Neither__ of them

__is__available to speak right now.

__Either__of us__is__capable of doing the job.### Rule 12

The words *here* and *there* have generally been labeled as adverbs even though they indicate place. In sentences beginning with *here* or *there*, the subject follows the verb.

**Examples:***There are four hurdles to jump.*

*There*

__is__a high__hurdle__to jump.### Rule 13

Use a singular verb with sums of money or periods of time.

**Examples:**

__Ten dollars____is__a high price to pay.

__Five years____is__the maximum sentence for that offense.### Rule 14

Sometimes the pronoun *who, that*, or *which* is the subject of a verb in the middle of the sentence. The pronouns *who, that*, and *which* become singular or plural according to the noun directly in front of them. So, if that noun is singular, use a singular verb. If it is plural, use a plural verb.

**Examples:***Salma is the scientist who writes/write the reports.*

The word in front of

*who*is

*scientist*, which is singular. Therefore, use the singular verb

*writes*.

*He is one of the men*

__who__does/__do__the work.The word in front of

*who*is

*men*, which is plural. Therefore, use the plural verb

*do.*

### Rule 15

Collective nouns such as *team* and *staff* may be either singular or plural depending on their use in the sentence.

**Examples:***The staff is in a meeting.*

*Staff*is acting as a unit here.

*The*

__staff____are__in disagreement about the findings.*The staff*are acting as separate individuals in this example.

The sentence would read even better as:

*The staff*

__members____are__in disagreement about the findings.Lovely Thread !