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Hence the remainder when 7 25 is divided by 9 is the remainder we obtain when the product 8 times 7 is divided by 9. The remainder is 7 in this case.
Hence the remainder when is divided by 9 is 7. What is the remainder when is divided by 7? Let Q 1 and R 1 be the quotient and the remainder when A is divided by B.
Hence to find the remainder when both the dividend and the divisor have a factor in common, Take out the common factor i. The real remainder R is this remainder R1 multiplied by the common factor k. What the remainder when 2 96 is divided by 96?
Removing 32 from the dividend and the divisor we get the numbers 2 91 and 3 respectively. The remainder when 2 91 is divided by 3 is 2. Hence the real remainder will be 2 multiplied by common factor Hence, the remainder when 15 is divided by 16 is 15 or 1. What is the remainder when is divided by 5?
Since the number is divisible by it will certainly be divisible by 5. Hence, the remainder is 0. Answer: The remainders when and are divided by 7 are 4 and 3 respectively. Hence, the problem reduces to finding the remainder when 4 3 is divided by 7. Now is divisible by 17 Theorem 3 and is divisible by 17 Theorem 2. Now is divisible by 19 Theorem 3 and is divisible by 19 Theorem 2. Hence the complete expression is also divisible by Answer: The remainder when the expression is divided by x 1 will be f 1.
Find the remainder when 5 37 is divided by What is the remainder when n 7 n is divided by 42?
Hence the remainder is 0. In other words, P 1! It also means that the remainder when P 1! Is divided by P is P 1 when P is prime.
Find the remainder when 40! Answer: By Wilson s theorem, we can see that 40! Therefore, 39! Chinese Remainder Theorem This is a very useful result. It might take a little time to understand and master Chinese remainder theorem completely but once understood, it is an asset. Following example will make it clear. Find the remainder when is divided by No confusion remains thereby. Answer: Let N be the number. The quotient of the division gives us the number of numbers divisible by p and less than or equal to n.
How many numbers less than are divisible by 12? Answer: Dividing by 12, we get the quotient as Hence the number of numbers that are below and divisible by 12 is How many numbers between 1 and , both included, are not divisible either by 3 or 5? Answer: We first find the numbers that are divisible by 3 or 5.
Dividing by 3 and 5, we get the quotients as and 80 respectively. Among these numbers divisible by 3 and 5, there are also numbers which are divisible both by 3 and 5 i. We have counted these numbers twice. Dividing by 15, we get the quotient as How many numbers between 1 and , both included, are not divisible by any of the numbers 2, 3 and 5? Answer: as in the previous example, we first find the number of numbers divisible by 2, 3, or 5. Some Special Problems: Find the remainder when is divided by Therefore, we first find the remainders when this number is divided by 9 and 4.
The remainder by 9 would be the remainder when the sum of digits is divided by 9. Therefore, to find the remainder we need to find the smallest multiple of 4 that gives remainder 1 with 9. The overall remainder would be the smallest number that gives remainder 3 with 9 and remainder 2 with 4. What is the number? Answer: Let the divisor be D and the remainder be R. The HCF of 50 and is Therefore, the highest number can be What is the remainder when is divided by ? This is left to students to check it out.
Answer: 0 values. If both are divisible by 11, their product is divisible by but 33 is divisible only by 11 therefore the expression is not divisible by If both are not divisible by 11, the expression is again not divisible by Example: The number 24 is divisible by 1, 2, 3, 4, 6, 8, 12, and Hence all these numbers are divisors of Similarly, any divisor of 60 will have powers of 3 equal to either 3 0 or 3 1, and powers of 5 equal to either 5 0 or 5 1.
To make a divisor of 60, we will have to choose a power of 2, a power of 3 and a power of 5. Similarly, a power of 3 can be chosen in 2 ways and a power of 5 can be chosen in 2 ways.
Notice that we have added 1 each to powers of 2, 3 and 5 and multiplied. How many divisors of are odd numbers? Answer: An odd number does not have a factor of 2 in it.
Therefore, we will consider all the divisors having powers of 3 and 5 but not 2. How many divisors of are even numbers? How many divisors of are not divisors of and how many divisors of are not divisors of ? Answer: The best option here is to find the number of common divisors of and For that we find the highest common powers of all the common prime factors in and Therefore, the two numbers will have 18 factors in common.
Answer: For unit digit equal to 5, the number has to be a multiple of 5 and it should not be a multiple of 2 otherwise the unit digit will be 0. To be a multiple of 5, the powers of 5 that it can have is 5 1, 5 2, 5 3 or 5 4. How many divisors of are perfect cubes?
Therefore, powers of 2 will be 2 0, 2 3, 2 6, 2 9, 2 72 and similarly, powers of 3 will be 3 0, 3 3, 3 6, 3 9, Both are 25 in number. Reverse Operations on Divisors: Find all the numbers less than which have exactly 8 divisors. Answer: To find the number of divisors of a number, we used to add 1 to powers of all the prime factors and then multiply them together.
Now, given the number of divisors, we will express this number as a product and then subtract 1 from every multiplicand to obtain the powers. Therefore, the number is of the form a 1 b 1 c 1, where a, b and c are prime. Therefore, the number is of the form a 3 b, where a and b are prime. The number can also be of the form a 7, but there is no such number less than Find the smallest number with 15 divisors.
To find the smallest such number, we give the highest power to smallest rime factor, i. Answer: In a perfect square, all the prime factors have even powers. The even powers of 2 are 2 0, 2 2, 2 4, even powers of 3 are 3 0 and 3 2, and even powers of 5 are 5 0 and 5 2. We can select an even power of 2 in 3 ways, even power of 3 in 2 ways, and even power of 5 in 2 ways.
Therefore, these common divisors will be multiplied only once. The common divisors will come from and are 36 in number. Therefore, divisors occur in pairs except for the square root for numbers which are perfect squares. Also, since we are asked for integers, the pair consisting of two negative integers will also suffice.
As the two factors will be prime to each other, we will have to assign a prime factor with its power for example completely to one of the factors. For every prime factor, we have two ways of assigning it.
The following examples will make it clear: How many of the first natural numbers are not divisible by any of 2, 3 and 5? Answer: is a multiple of 2, 3 and 5. Answer: Unlike the previous problem, this problem only asks for number not divisible by only 2 factors of , i. Therefore, in the formula we remove the part containing the factor of 3 and calculate the numbers of numbers prime to with respect to prime factors 2 and 5.
These are in numbers in all we have already calculated it. Which natural number has the highest number of divisors? To calculate units digit of we only consider the units digit of Hence, we find the units digit of To find the units digit of a b, we only consider the units digits of the numbers a and b. To calculate units digit of , we only consider the units digit of and i. Hence, we find the units digit of 3 x 4, respectively. To calculate units digit of x y where x is a single digit number To calculate units digit of numbers in the form x y such 7 , 8 93, 3 74 etc.
This book is probably the best possible way to develop a good vocabulary. The problem, however, with this book is that it is voluminous and bulky. Rarely have I come across a CAT aspirant who has been able to stick to this book for over a month.
They start it with full gusto but give up on it in the middle. If building vocabulary is your focus, using some mobile apps is a good idea. Here is a list of recommended books for CAT aspirants. It is more of a workbook and has a bunch of exercises that can help you improve your reading speed. But perhaps — that is not something that you need. If you have a good reading speed, it helps.
But probably it should not be your priority. The length of CAT passages have ranged from words to words in the past decade. You can also say that it is anywhere from words per question. Questions in CAT have rarely been so simple that if you speed-read the passage, you will be able to answer them.
You might not need a deep level understanding of the passage but a superficial glance is not going to help you either. You might improve a little bit if you go through the exercises mentioned in the book but I would still recommend that you improve your reading habits rather than improving your reading speed.
NCERT Books — I have been guilty of recommending the same to a few students who struggle with the basics of a particular topic. Over time I have realised why this recommendation does not work. The reason is not that these books focus on the wrong concepts. The reason is that they are not required. If you as a student are unable to understand a particular concept, it is not because of the book you are using but it is because you are not able to understand what is given in the book.
Probably you need someone to explain it to you. You need someone who can clarify your doubts. The book is not at fault here — you are!
I am sorry if this sounds rude. You should stick to your CAT books and study material. High School English Grammar and Composition by Wren and Martin — It is a great book but as the name suggests, you should have read it in high school.
The importance of Grammar, sadly, has reduced over the years in CAT. Even earlier, CAT did not mean to test if you knew the definition of a gerund or how it is different from an infinitive.
The objective always was to check if you were able to identify the correct sentence or not.