The most important topics in Number Systems for CAT are divisibility, factors, cyclicity of factors, remainder theorems, highest power of a number in a factorial, and the last few digits of a number. All these topics are covered in the questions given below which come with detailed explanations and video solutions.

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Instructions

For the following questions answer them individually

Question 1

Let ‘a’ and ‘b’ be two 2 digit numbers such that b is obtained by reversing the digits of a. It is also known that they satisfy the relation

$$a^{2} - b^{2} = k^{2}$$ where k is some positive integer. What is the value of a+b+k?

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123

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789

456

123

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Question 2

X is the smallest number which leaves a remainder of 2 when divided by 7 and a remainder of 1 when divided by 19. What is the remainder when X is divided by 23?

Question 3

How many pairs of integers exist such that the difference between their product and sum is 72?

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Question 4

$$(245)_{x}+(162)_{x}-(427)_{x}=0$$ in some base x. What is the value of x?

Question 5

Find the remainder when 16! + 86 is divided by 323

Question 6

Find the number of natural number pairs ( a,b ) such that 7*a - 20*b = 1 and a < 2000.

Question 7

How many even factors of 135000 are not factors of 10800?

Question 8

How many ordered quadruplets (x,y,z,w) are possible such that x!+y!+z! = $$3^w$$. Given that x, y, z, w are natural numbers such that x>y>z.

Question 9

If the units digit of $$1^{1!}+2^{2!}+ 3^{3!}+ 4^{4!}+ 5^{5!}+….. (k-1)^{k-1!}$$ is same as that of $$1^{1!}+2^{2!}+ 3^{3!}+ 4^{4!}+ 5^{5!}+….. k^{k!}$$. Given that $$k$$ is a natural number less than or equal to 105, how many values can k assume?

Question 10

If $$17 < a < 67$$, how many integral pairs of (a, b) are possible satisfying the equation $$\frac{3}{a} + \frac{2}{b} = \frac{1}{18}$$?

Question 11

A book contains 57 stories each with fewer than 58 pages. The first story starts on the 2nd page and each story thereafter, starts on a fresh page. What is the largest number of stories than can begin on an odd numbered page?

Question 12

How many distinct natural numbers 'n' are there such that, amongst all its divisors, greater than 1 and less than 'n', the largest divisor is 21 times the smallest divisor?

Question 13

If X and Y are multiples of 7 with X>Y>0, which of the following statements is not always true?

Question 14

What is the remainder when $$16^3$$ + $$17^3$$ + $$18^3$$ + $$19^3$$ is divided by 70 ?

Question 15

A six - digit number N is formed using the digits 0, 3, 6 and 9 only. Each of the digits is used at least once. It was found that N is divisible by 18. What is the ten’s digit of the smallest such six-digit number?

Question 16

a,b,c,d are four integers such that a>b>c>d. The largest integer which will always divide (a-b)(a-c)(a-d)(b-c)(b-d)(c-d) is

Question 17

How many number $$X$$ less than 350 exist such that the sum of the number of divisors of $$X$$ and $$X^{2}$$ is 60?

Question 18

At a printing press, the machine develops some problem. Due to this problem the machine skips the numbers which contain digits 5 and 7 and instead prints the next number(For example after 49, it will print 60). However, since it is urgent the owner goes ahead with printing a novel. Ajay and Vijay buy the same novel but Vijay’s novel was printed at this faulty printing press. Ajay describes some scene of the novel to Vijay which is there on page number 189 in Ajay’s book. Vijay’s checks his novel but cannot find it on page 189 in his book. On which page should he look in his book, so that he can find the scene which was described by Ajay?

Question 19

For all natural numbers n, suppose f(n) = n + sum of the digits of n. Then, for how many natural numbers n is f(n) = 120 ?

Question 20

Find the remainder when $$7^{21}+49^{21}+343^{21}+2401^{21}$$ is divided by $$7^{20}+1$$.

Question 21

$$(245)_{x}+(162)_{x}-(427)_{x}=0$$ in some base x. What is the value of x?

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