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.
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).
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).
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
104^303 = (3 mod 101)^3 * 1
= 27 ANSWER :)
9^1 +9^2 +..............+9^8
is divided by 6?
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
which is div by 6
@rudra13 : thanks ...
y=1/3 +3/18+15/162+........... find the value of y^2+2y.
right ans is 2
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 )
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 :
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.
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.
Two singular subjects connected by or or nor require a singular verb.
My aunt or my uncle is arriving by train today.
Two singular subjects connected by either/or or neither/nor require a singular verb as in Rule 1.
Neither Juan nor Carmen is available.
Either Kiana or Casey is helping today with stage decorations.
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.
Neither she nor I am going to the festival.
When a singular subject is connected by or or nor to a plural subject, put the plural subject last and use a plural verb.
The serving bowl or the plates go on that shelf.
When a singular and plural subject are connected by either/or or neither/nor, put the plural subject last and use a plural verb.
Neither Jenny nor the others are available.
As a general rule, use a plural verb with two or more subjects when they are connected by and.
A car and a bike are my means of transportation.
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.
The politician, along with the newsmen, is expected shortly.
Excitement, as well as nervousness, is the cause of her shaking.
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.
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.
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.
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).
The expression the number is followed by a singular verb while the expression a number is followed by a plural verb.
The number of people we need to hire is thirteen.
A number of people have written in about this subject.
When either and neither are subjects, they always take singular verbs.
Neither of them is available to speak right now.
Either of us is capable of doing the job.
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.
There are four hurdles to jump.
There is a high hurdle to jump.
Use a singular verb with sums of money or periods of time.
Ten dollars is a high price to pay.
Five years is the maximum sentence for that offense.
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.
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.
Collective nouns such as team and staff may be either singular or plural depending on their use in the sentence.
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 !