# 100+ CAT Number Systems Questions With Video Solutions

## CAT Number Systems Questions Weightage

 Year Weightage 2023 7 2022 4 2021 2 2020 9 2019 5 2018 4

## CAT Number Systems Formulas PDF

CAT Number systems is one of the most important topics in the quantitative aptitude section, and it is vital to have a clear understanding of the formulas related to them. To help the aspirants to ace this topic, we have made a PDF containing a comprehensive list of formulas, tips, and tricks that you can use to solve number systems questions with ease and speed. Click on the below link to download the CAT Number Systems formulas PDF.

1. Remainder Theorems Formulae

Fermat's Theorem - For any integer $$a$$ and prime number $$p$$, $$a^p-a$$ is always divisible by $$p$$

Wilson's Theorem - For a prime $$p$$, remainder when $$(p-1)!$$ i divided by $$p$$ is $$(p-1)$$

Euler's Theorem - If M and N are co-prime to each other then the remainder when $$M^{\phi(N)}$$ is divided by N is 1

2. HCF and LCM

HCF * LCM of two numbers = Product of two numbers

The greatest number dividing a, b and c leaving remainders of $$x_1$$, $$x_2$$ and $$x_3$$ is the HCF of (a-$$x_1$$), (b-$$x_2$$) and (c-$$x_3$$).

The greatest number dividing a, b and c (a<b<c) leaving the same remainder each time is the HCF of (c-b), (c-a), (b-a).

LCM of fractions = LCM of Numerators ÷ HCF of Denominators.

3. Number of trailing zeros

Number of trailing zeros of n! in base b(b=$$p^m$$, where p is a prime number) is for $$k\ge1$$ $$\frac{1}{m}\left(\Sigma\left[\frac{n}{p^k}\right]\ \right)$$

## CAT 2023 Number Systems questions

#### Question 1

Let a, b, m and n be natural numbers such that $$a>1$$ and $$b>1$$. If $$a^{m}b^{n}=144^{145}$$, then the largest possible value of $$n-m$$ is

#### Question 2

Let n be the least positive integer such that 168 is a factor of $$1134^{n}$$. If m is the least positive integer such that $$1134^{n}$$ is a factor of $$168^{m}$$, then m + n equals

#### Question 3

For any natural numbers m, n, and k, such that k divides both $$m+2n$$ and $$3m+4n$$, k must be a common divisor of

#### Question 4

The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is

#### Question 5

The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

#### Question 6

The number of coins collected per week by two coin-collectors A and B are in the ratio 3 : 4. If the total number of coins collected by A in 5 weeks is a multiple of 7, and the total number of coins collected by B in 3 weeks is a multiple of 24, then the minimum possible number of coins collected by A in one week is

#### Question 7

The number of all natural numbers up to 1000 with non-repeating digits is

## CAT 2022 Number Systems questions

#### Question 1

Consider six distinct natural numbers such that the average of the two smallest numbers is 14, and the average of the two largest numbers is 28. Then, the maximum possible value of the average of these six numbers is

#### Question 2

Let A be the largest positive integer that divides all the numbers of the form $$3^k + 4^k + 5^k$$, and B be the largest positive integer that divides all the numbers of the form $$4^k + 3(4^k) + 4^{k + 2}$$ , where k is any positive integer. Then (A + B) equals

#### Question 3

For some natural number n, assume that (15,000)! is divisible by (n!)!. The largest possible value of n is

#### Question 4

A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the students in the school is

## CAT 2021 Number Systems questions

#### Question 1

For all possible integers n satisfying $$2.25\leq2+2^{n+2}\leq202$$, then the number of integer values of $$3+3^{n+1}$$ is:

#### Question 2

For a 4-digit number, the sum of its digits in the thousands, hundreds and tens places is 14, the sum of its digits in the hundreds, tens and units places is 15, and the tens place digit is 4 more than the units place digit. Then the highest possible 4-digit number satisfying the above conditions is

## CAT 2020 Number Systems questions

#### Question 1

How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?

#### Question 2

How many 4-digit numbers, each greater than 1000 and each having all four digits distinct, are there with 7 coming before 3?

#### Question 3

Let m and n be natural numbers such that n is even and $$0.2<\frac{m}{20},\frac{n}{m},\frac{n}{11}<0.5$$. Then $$m-2n$$ equals

#### Question 4

How many integers in the set {100, 101, 102, ..., 999} have at least one digit repeated?

#### Question 5

Let N, x and y be positive integers such that $$N=x+y,2<x<10$$ and $$14<y<23$$. If $$N>25$$, then how many distinct values are possible for N?

#### Question 6

How many of the integers 1, 2, … , 120, are divisible by none of 2, 5 and 7?

#### Question 7

How many pairs(a, b) of positive integers are there such that $$a\leq b$$ and $$ab=4^{2017}$$ ?

### How can video solutions help me in preparing for CAT Number Systems questions?

Video solutions can be a helpful resource for candidates preparing for CAT Number Systems questions. They can provide a step-by-step explanation of how to solve the problem, helping candidates better understand the concept and formula. Also, one can find various tips, tricks and shortcuts to solve the questions quickly.

### Is the number system easy for CAT?

Usually, the questions in the CAT from Number systems are moderately tricky. But not so tough if you are well versed with the basics and practice a good number of questions from this topic.