Number System CAT PYQs , 107 Questions with Solutions

Let $3 \leq x \leq 6$ and $\lfloor x^2 \rfloor = \lfloor x \rfloor^2$, where $\lfloor x \rfloor$ is the greatest integer not exceeding $x$. If set $S$ represents all feasible values of $x$, then a possible subset of $S$ is:

In a 3-digit number N, the digits are non-zero and distinct such that none of the digits is a perfect square, and only one of the digits is a prime number. Then, the number of factors of the minimum possible value of N is

The number of divisors of $\left(2^6 \times 3^5 \times 5^3 \times 7^2\right)$, which are of the form $(3r+1)$, where $r$ is a non-negative integer, is

Suppose $a , b , c$ are three distinct natural numbers, such that $3 a c = 8 ( a + b )$. Then, the smallest possible value of $3 a + 2 b + c$ is

What value comes in place of the question mark (?) in the following statements, and arrange them in descending order.

A. $\frac{5}{9} \times (225.40 - 45.4) = (?)^2$

B. $\sqrt{1024} \times 40 + (20)^2 + 0.5\% \text{ of } 9600 + 469 = (?)^3$

C. $8^{12} \div 16^2 \text{ of } 32^3 \times \sqrt{256} = (2)^?$

D. $18\frac{1}{3} + 9\frac{2}{3}: 10\frac{1}{3} = 1\frac{2}{3} + ?$

Choose the correct answer from the options given below:

If $m$ and $n$ are integers such that $( m + 2 n ) ( 2 m + n ) = 27$, then the maximum possible value of $2 m - 3 n$ is

For a 4-digit number (greater than 1000), sum of the digits in the thousands, hundreds, and tens places is 15. Sum of the digits in the hundreds, tens, and units places is 16. Also, the digit in the tens place is 6 more than the digit in the units place. The difference between the largest and smallest possible value of the number is

The sum of all the digits of the number $\left( 10 ^ { 50 } + 10 ^ { 25 } - 123 \right)$, is

Arun selected an integer x between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer x and any other integer between 2 and 40, both inclusive, is 1. How many different choices for such an x are possible?

If a, b, and c are all positive integers, with 4a > b, then which of the following conditions is BOTH NECESSARY AND SUFFICIENT for the expression $\sqrt[3]{3^a \cdot 21^{3a-b} \cdot 49^{2b+c}}$ to be a positive integer?

When $10^{100}$ is divided by 7 , the remainder is

For any natural number $n$ , let $a_n$ be the largest integer not exceeding $\sqrt{n}$ . Then the value of $a_1+a_2+\cdots+a_{50}$ is

If the digit in the unit's place of a two-digit number is halved and the digit in ten's place is doubled, the number thus obtained is equal to the number obtained by interchanging the digits. Which of the following is definitely true?

If abc > 0, such that a, b, c are integers then which of the following must be true?

If $m$ and $n$ are natural numbers such that $n>1$, and $m^n=2^{25} \times 3^{40}$, then $m-n$ equals

When $3^{333}$ is divided by 11 , the remainder is

If $10^{68}$ is divided by 13 , the remainder is

The least common multiple of a number and 990 is 6930. The greatest common divisor of that number and 550 is 110.

What is the sum of the digits of the least possible value of that number?

Consider a 4-digit number of the form $abbb$, i.e., the first digit is $a\ (a > 0)$ and the last three digits are all $b$. Which of the following conditions is both NECESSARY and SUFFICIENT to ensure that the 4-digit number is divisible by $a$?

There are 3 different types of rice weighing 435,493 and 551 kg respectively. The rice is packed in bags, such that no two types of rice are mixed. Find the least no of bags of equal sizes.

Find the missing number?

(A number pyramid with 4 rows. Bottom row: 5, 3, 6, 4. Third row: 5, 8, ?. Second row: 30, 102. Top row: 3050.)

A king has distributed all his rare jewels in three boxes. The first box contains $\dfrac{1}{3}$ of the rare jewels, while the second box contains $\dfrac{k}{5}$ of the rare jewels, for some positive integer value of $k$. The third box contains 66 rare jewels. How many rare jewels does the king have?

Find the smallest number between 2000 and 3000 which is exactly divisible by 21,24 and 28

There are two numbers which are greater than 21 and their LCM and HCF are 3003 and 21 respectively. What is the sum of these numbers?

Suppose Haruka has a special key Δ in her calculator called delta key: Rule 1: If the display shows a one-digit number, pressing delta key Δ replaces the displayed number with twice its value. Rule 2: If the display shows a two-digit number, pressing delta key Δ replaces the displayed number with the number equal to the sum of its digits. Suppose Haruka enters the value 1 and then presses delta key Δ repeatedly. After pressing the key for 68 times, what will be the displayed number?

Amit has forgotten his 4-digit locker key. He remembers that all the digits are positive integers and are different from each other. Moreover, the fourth digit is the smallest and the maximum value of the first digit is 3. Also, he recalls that if he divides the second digit by the third digit, he gets the first digit. How many different combinations does Amit have to try for unlocking the locker?

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

The addition of 7 distinct positive integers is 1740. What is the largest possible “greatest common divisor” of these 7 distinct positive integers?

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

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

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

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

A flower bed has an area of 125, 375 and 250 square meters. What is the maximum rows of flowers a farmer can grow if the width of a row is 3 square meters.

3 + 2 = 7, 51 + ? = 76, 32 - 8 = -32, 12 + 3 = 21 Find the value of ?

Raju and Sarita play a number game. First, each one of them chooses a positive integer independently. Separately, they both multiply their chosen integers by 2, and then subtract 20 from their resultant numbers. Now, each of them has a new number. Then, they divide their respective new numbers by 5. Finally, they added their results and found that the sum is 16. What can be the maximum possible difference between the positive integers chosen by Raju and Sarita?

Let $x$ and $y$ be two positive integers and $p$ be a prime number. If $x(x-p)-y(y+p)=7p$, what will be the minimum value of $x-y$?

The sum of the cubes of two numbers is 128, while the sum of the reciprocals of their cubes is 2. What is the product of the squares of the numbers?

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 $4^k+3\left(4^k\right)+4^{k+2}$ , where $k$ is any positive integer. Then $(A+B)$ equals

Wilma, Xavier, Yaska and Zakir are four young friends, who have a passion for integers. One day, each of them selects one integer and writes it on a wall. The writing on the wall shows that Xavier and Zakir picked positive integers, Yaska picked a negative one, while Wilma’s integer is either negative, zero or positive. If their integers are denoted by the first letters of their respective names, the following is true: $W^4 + X^3 + Y^2 + Z \le 4$ $X^3 + Z \ge 2$ $W^4 + Y^2 \le 2$ $Y^2 + Z \ge 3$ Given the above, which of these can $W^2 + X^2 + Y^2 + Z^2$ possibly evaluate to?

For natural numbers $x$, $y$, and $z$, if $xy + yz = 19$ and $yz + xz = 51$, then what is the minimum possible value of $xyz$?

Nadeem’s age is a two-digit number X, squaring which yields a three-digit number, whose last digit is Y. Consider the statements below: Statement I: Y is a prime number. Statement II: Y is one-third of X. To determine Nadeem’s age uniquely:

For any real number $x$ , let $[x]$ be the largest integer less than or equal to $x$ . If $\sum_{n=1}^N\left[\frac{1}{5}+\frac{n}{25}\right]=25$ then $N$ is

A supplier receives orders from 5 different buyers. Each buyer places their order only on a Monday. The rst buyer places the order after every 2 weeks, the second buyer, after every 6 weeks, the third buyer, after every 8 weeks, the fourth buyer, every 4 weeks, and the fth buyer, after every 3 weeks. It is known that on January 1st, which was a Monday, each of these ve buyers placed an order with the supplier. On how many occasions, in the same year, will these buyers place their orders together excluding the order placed on January 1st?

Fatima found that the profit earned by the Bala dosa stall today is a three-digit number. She also noticed that the middle digit is half of the leftmost digit, while the rightmost digit is three times the middle digit. She then randomly interchanged the digits and obtained a different number. This number was more than the original number by 198. What was the middle digit of the profit amount?

Shireen draws a circle in her courtyard. She then measures the circle’s circumference and its diameter with her measuring tape and records them as two integers, A and B respectively. She finds that A and B are co-primes, that is, their greatest common divisor is 1. She also finds their ratio, A:B, to be: 3.141614161416… (repeating endlessly). What is A - B?

The Madhura Fruits Company is packing four types of fruits into boxes. There are 126 oranges, 162 apples, 198 guavas and 306 pears. The fruits must be packed in such a way that a given box must have only one type of fruit and must contain the same number of fruit units as any other box. What is the minimum number of boxes that must be used?

The number of integers greater than 2000 that can be formed with the digits 0, 1, 2, 3, 4, 5, using each digit at most once, is

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

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

If 1/x is a positive fraction and 1/y is a negative fraction then which of the following statements are true?

(A) 1/x + 1/y is positive (B) 1/x − 1/y is positive (C) (x − y)/xy is negative (D) 1/xy is positive (E) 1/x² − 1/y² is positive

Choose the correct answer from the options given below:

The sum of all natural numbers less than 201 which are divisible by either 5 or 6 but not by both is?

A two digit number increases by 75% when it's digits get reversed. Find the original number when the difference between the digits of the two-digit number is 3?

The largest 4 digit number that leaves the same remainder when divided by 2, 3, 4 and 5 is

Arrange the following rational numbers in ascending order: (A) −4/5 (B) −5/12 (C) −7/18 (D) −2/3

How many three-digit numbers are greater than 100 and increase by 198 when the three digits are arranged in the reverse order?

M is a 4-digit number. If the left most digit is removed, then the resulting three digit number is 1/9th of M. How many such M's are possible?

The dimensions of a floor are 18 × 24. What is the smallest number of identical square tiles that pave the entire floor without the need to break any tile?

A box has 450 balls, each either white or black, there being as many metallic white balls as metallic black balls. If 40% of the white balls and 50% of the black balls are metallic, then the number of non-metallic balls in the box is

Given below are two statements Statement I: (543) base 6 is equivalent to (317) base 8 Statement II: The last 4 bits in the binary representation of a multiple of 16 is 1000. In light of the above statements, choose the correct answer from the options given below

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

In a tournament, a team has played 40 matches so far and won 30% of them. If they win 60% of the remaining matches, their overall win percentage will be 50%. Suppose they win 90% of the remaining matches, then the total number of matches won by the team in the tournament will be

Zahir and Raman are at the entrance of a dark cave. To enter this cave, they need to open a number lock. Raman sees a note on a rock: “... chest of pure diamonds kept for the smart one ... number has six digits ... second last digit is 2, third last is 4 ... divisible by all prime numbers less than 15 ...”. Excited, Zahir and Raman seek your help: which of these can be the rst digit of the six-digit number that will help them open the lock?

Symbiosis runs a corporate training programme. At the end of the first programme, the total takings were Rs. 38950. There were more than 45 but fewer than 100 participants. What was the participants' fee?

The difference between two numbers is 1365. When the larger is divided by the smaller, the quotient is 6 and the remainder is 15. What is the smaller number?

When a heap of pebbles is grouped in 32, 40, or 72, it leaves remainders 10, 18, and 50 respectively. What is the minimum number of pebbles in the heap?

If n = 1 + X where X is the product of four consecutive positive integers, which of the following are true?

1. n is odd
2. n is prime
3. n is a perfect square.

If a, b and c are positive integers such that ab = 432, bc = 96 and c < 9, then the smallest possible value of a + b + c is

Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is

The mean of all 4 digit even natural numbers of the form 'aabb', where a>0, is

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

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

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

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 $N$?

How many pairs (a,b) of positive integers are there such that a ≤ b and ab = 42017 ?

What is the remainder if $19^{20} - 20^{19}$ is divided by $7$?

When expressed in a decimal form, which of the following numbers will be non-terminating as well as non-repeating?

How many factors of $2^4 \times 3^5 \times 10^4$ are perfect squares which are greater than 1? [TITA]

In a six-digit number, the sixth, that is, the rightmost, digit is the sum of the first three digits, the fifth digit is the sum of first two digits, the third digit is equal to the first digit, the second digit is twice the first digit and the fourth digit is the sum of fifth and sixth digits. Then, the largest possible value of the fourth digit is [TITA]

How many pairs $(m,n)$ of positive integers satisfy the equation $m^2 + 105 = n^2$?

Rahul asked Shyam to find N such that N! > 10^N and must be the smallest integer. Shyam calculates and find N is between 10 to 15, but he asks his friend Sohan who says it is between 16 to 20. Then, Shyam asks his neighbour Suresh who says it is between 21 to 25. Then, he asks his cousin Sonal who says it is between 26 to 31. Who is correct?

Find the unit's place digit of (1!)^1! + (2!)^2! + (3!)^3! + ... + (100!)^100!

Find the number of zeros at the end of (5!)^5! + (10!)^10! + (15!)^15! + ... + (100!)^100!

We have two unknown positive integers $m$ and $n$, whose product is less than $100$.

There are two additional statement of facts available:

$1)$ $mn$ is divisible by six consecutive integers $\{j,\, j+1,\,\ldots,\, j+5\}$

$2)$ $m+n$ is a perfect square.

Which of the two statements above, alone or in combination shall be sufficient to determine the numbers $m$ and $n$?

What is the sum of integers from 113 to 113113, which are divisible by 7?

The number of integers $x$ such that $0.25 < 2^x < 200$, and $2^x + 2$ is perfectly divisible by either 3 or 4, is [TITA]

While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is: [TITA]

The smallest integer $n$ for which $4^n > 17^{19}$ holds, is closest to

If $N$ and $x$ are positive integers such that $N^N = 2^{160}$ and $N^2 + 2^N$ is an integral multiple of $2^x$, then the largest possible value of $x$ is

If the sum of squares of two numbers is 97, then which one of the following cannot be their product?

How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?

What will be the numerical value of MMXVIII

$X$ and $Y$ are the digits at the unit's place of the numbers $(408X)$ and $(789Y)$ where $X \ne Y$. However, the digits at the unit's place of the numbers $(408X)^{63}$ and $(789Y)^{85}$ are the same. What will be the possible value(s) of $(X + Y)$?

Find the value of the expression:

$10 + 10^3 + 10^6 + 10^9$

The number of solutions (x, y, z) to the equation x – y – z = 25, where x, y, and z are positive integers such that x ≤ 40, y ≤ 12, and z ≤ 12 is

If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a - b)² + (a - c)² + (a - d)² is (TITA)

An elevator has a weight limit of 630 kg. It is carrying a group of people of whom the heaviest weighs 57 kg and the lightest weighs 53 kg. What is the maximum possible number of people in the group? [TITA]

Let $a_1, a_2, a_3, a_4, a_5$ be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers, ending with $2a_3$. If the sum of the numbers in the new sequence is $450$, find $a_5$.

If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is

How many different pairs $(a, b)$ of positive integers are there such that $a \leq b$ and $\frac{1}{a} + \frac{1}{b} = \frac{1}{9}$?

What numbers should replace the two question marks in the grid below?

How many factors of 2^8 × 3^6 × 5^4 × 10^5 are multiples of 120?

If $N=(11^{p+7})(7^{q-2})(5^{r+1})(3^s)$ is a perfect cube, where $p,q,r$ and $s$ are positive integers, then the smallest value of $p+q+r+s$ is:

What is the value of (p-1)^(q) [with an additional +1, per the source notation], if pq=169 and p and q are whole numbers?

Hari’s family consisted of his younger brother (Chari), younger sister (Gouri), and their father and mother. When Chari was born, the sum of the ages of Hari, his father and mother was 70 years. The sum of the ages of four family members, at the time of Gouri’s birth, was twice the sum of ages of Hari’s father and mother at the time of Hari’s birth. If Chari is 4 years older than Gouri, then find the difference in age between Hari and Chari.

Kiran, Kumar and Kedar were watching cricket and eating groundnuts. They pop a groundnut in their mouth every 40 seconds, 72 seconds and 108 seconds respectively and munch it. If all of them pop a groundnut simultaneously at 8:20:00 hours, then at what time will they again pop groundnuts again in their mouth simultaneously?

The remainder obtained when 2^97 is divided by 97 is

Find the odd one in the series: 10, 16, 24, 28, 30