Top 130 CAT Number Systems Questions With Solutions

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

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

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

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:

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

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

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

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

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

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

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

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

What is the largest positive integer n such that $$\frac{n^2 + 7n + 12}{n^2 - n - 12}$$ is also a positive integer?

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

The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157:3, then the sum of the two numbers is

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

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

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

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

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

The number of integers x such that $$0.25 \leq 2^x \leq 200$$ and $$2^x + 2$$ is perfectly divisible by either 3 or 4, is

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

Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say m, of these three integers?

The integers 1, 2, …, 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b - 1 is written. What will be the number left on the board at the end?

What are the last two digits of $$7^{2008}$$?

A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of x?

How many pairs of positive integers m, n satisfy 1/m + 4/n = 1/12 , where n is an odd integer less than 60?

In a tournament, there are n teams $$T_1 , T_2 ....., T_n$$ with $$n > 5$$. Each team consists of k players, $$k > 3$$. The following pairs of teams have one player in common: $$T_1$$ & $$T_2$$ , $$T_2$$ & $$T_3$$ ,......, $$T_{n-1}$$ & $$T_n$$ , and $$T_n$$ & $$T_1$$ . No other pair of teams has any player in common. How many players are participating in the tournament, considering all the n teams together?

Consider four digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?

What are the values of x and y that satisfy both the equations?

$$2^{0.7x} * 3^{-1.25y} = 8\sqrt{6}/27$$

$$4^{0.3x} * 9^{0.2y} = 8*81^{1/5}$$

The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can possibly be one of these four numbers?

The number of employees in Obelix Menhir Co. is a prime number and is less than 300. The ratio of the number of employees who are graduates and above, to that of employees who are not, can possibly be:

If a/b = 1/3, b/c = 2, c/d = 1/2 , d/e = 3 and e/f = 1/4, then what is the value of abc/def ?

Let $$n!=1*2*3* ...*n$$ for integer $$n \geq 1$$.

If $$p = 1!+(2*2!)+(3*3!)+... +(10*10!)$$, then $$p+2$$ when divided by 11! leaves a remainder of

The digits of a three-digit number A are written in the reverse order to form another three-digit number B. If B > A and B-A is perfectly divisible by 7, then which of the following is necessarily true?

The rightmost non-zero digit of the number $$30^{2720}$$ is

For a positive integer n, let $$P_n$$ denote the product of the digits of n, and $$S_n$$ denote the sum of the digits of n. The number of integers between 10 and 1000 for which $$P_n$$ + $$S_n$$ = n is

Let S be a set of positive integers such that every element n of S satisfies the conditions

A. 1000 <= n <= 1200

B. every digit in n is odd

Then how many elements of S are divisible by 3?

Let $$x = \sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4- \ to \ infinity}}}}$$. Then x equals

If R = $$(30^{65}-29^{65})/(30^{64}+29^{64})$$ ,then

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is

If x = $$(16^3 + 17^3+ 18^3+ 19^3 )$$, then x divided by 70 leaves a remainder of

On January 1, 2004 two new societies S1 and S2 are formed, each n numbers. On the first day of each subsequent month, S1 adds b members while S2 multiples its current numbers by a constant factor r. Both the societies have the same number of members on July 2, 2004. If b = 10.5n, what is the value of r?

Suppose n is an integer such that the sum of digits on n is 2, and $$10^{10} < n < 10^{11}$$. The number of different values of n is

Each family in a locality has at most two adults, and no family has fewer than 3 children.

Considering all the families together, there are adults than boys, more boys than girls, and more girls than families.

Then the minimum possible number of families in the locality is

The remainder, when $$(15^{23} + 23^{23})$$ is divided by 19, is

In NutsAndBolts factory, one machine produces only nuts at the rate of 100 nuts per minute and needs to be cleaned for 5 minutes after production of every 1000 nuts.

Another machine produces only bolts at the rate of 75 bolts per minute and needs to be cleaned for 10 minutes after production of every 1500 bolts. If both the machines start production at the same time, what is the minimum duration required for producing 9000 pairs of nuts and bolts?

The total number of integers pairs (x, y) satisfying the equation x + y = xy is

The number of non-negative real roots of $$2^x - x - 1 = 0$$ equals

Twenty-seven persons attend a party. Which one of the following statements can never be true?

How many even integers n, where $$100 \leq n \leq 200$$ , are divisible neither by seven nor by nine?

A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of the three cases the leading digit is 1. Then M equals

How many three digit positive integers, with digits x, y and z in the hundred's, ten's and unit's place respectively, exist such that x < y, z < y and x $$\neq$$ 0 ?

There are 8436 steel balls, each with a radius of 1 centimeter, stacked in a pile, with 1 ball on top, 3 balls in the second layer, 6 in the third layer, 10 in the fourth, and so on. The number of horizontal layers in the pile is

In a certain examination paper, there are n questions. For j = 1,2 …n, there are $$2^{n-j}$$ students who answered j or more questions wrongly. If the total number of wrong answers is 4095, then the value of n is

The number of positive integers n in the range $$12 \leq n \leq 40$$ such that the product (n -1)*(n - 2)*…*3*2*1 is not divisible by n is

Let T be the set of integers {3,11,19,27,…451,459,467} and S be a subset of T such that the sum of no two elements of S is 470. The maximum possible number of elements in S is

Number S is obtained by squaring the sum of digits of a two-digit number D. If difference between S and D is 27, then the two-digit number D is

A rich merchant had collected many gold coins. He did not want anybody to know about him. One day, his wife asked, " How many gold coins do we have?" After a brief pause, he replied, "Well! if I divide the coins into two unequal numbers, then 48 times the difference between the two numbers equals the difference between the squares of the two numbers." The wife looked puzzled. Can you help the merchant's wife by finding out how many gold coins the merchant has?

When $$2^{256}$$ is divided by 17, the remainder would be

At a bookstore, ‘MODERN BOOK STORE’ is flashed using neon lights. The words are individually flashed at the intervals of 2.5 s, 4.25 s and 5.125 s respectively, and each word is put off after a second. The least time after which the full name of the bookstore can be read again for a full second is

Three pieces of cakes of weights 4.5 lb, 6.75 lb and 7.2 lb respectively are to be divided into parts of equal weight. Further, each part must be as heavy as possible. If one such part is served to each guest, then what is the maximum number of guests that could be entertained?

After the division of a number successively by 3, 4 and 7, the remainders obtained are 2, 1 and 4 respectively. What will be the remainder if 84 divides the same number?

$$7^{6n} - 6^{6n}$$, where n is an integer > 0, is divisible by

Two boys are playing on a ground. Both the boys are less than 10 years old. Age of the younger boy is equal to the cube root of the product of the age of the two boys. If we place the digit representing the age of the younger boy to the left of the digit representing the age of the elder boy, we get the age of father of the younger boy. Similarly, if we place the digit representing the age of the elder boy to the left of the digit representing the age of the younger boy and divide the figure by 2, we get the age of mother of the younger boy. The mother of the younger boy is younger to his father by 3 years. Then, what is the age of the younger boy?

Of 128 boxes of oranges, each box contains at least 120 and at most 144 oranges. X is the maximum number of boxes containing the same number of oranges. What is the minimum value of X?

In a four-digit number, the sum of the first 2 digits is equal to that of the last 2 digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other 2 digits. What is the third digit of the number?

Anita had to do a multiplication. In stead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?

In a number system the product of 44$$_{10}$$ and 11$$_{10}$$ is 3414. The number 3111 of this system, when converted to the decimal number system, becomes

All the page numbers from a book are added, beginning at page 1. However, one page number was added twice by mistake. The sum obtained was 1000. Which page number was added twice?

A set of consecutive positive integers beginning with 1 is written on the blackboard. A student came along and erased one number. The average of the remaining numbers is $$\frac{602}{17}$$. What was the number erased?

Let $$b$$ be a positive integer and $$a = b^2 - b$$. If $$b \geq 4$$ , then $$a^2 - 2a$$ is divisible by

Ashish is given Rs. 158 in one-rupee denominations. He has been asked to allocate them into a number of bags such that any amount required between Re 1 and Rs. 158 can be given by handing out a certain number of bags without opening them. What is the minimum number of bags required?

In some code, letters a, b, c, d and e represent numbers 2, 4, 5, 6 and 10. We just do not know which letter represents which number. Consider the following relationships:

I. a + c = e,
II. b - d = d and
III. e + a = b

Which of the following statements is true?

Let n be the number of different five-digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?

Let x, y and z be distinct integers. x and y are odd and positive, and z is even and positive. Which one of the following statements cannot be true?

A red light flashes three times per minute and a green light flashes five times in 2 min at regular intervals. If both lights start flashing at the same time, how many times do they flash together in each hour?

Let D be recurring decimal of the form, $$D = 0.a_1a_2a_1a_2a_1a_2...$$, where digits $$a_1$$ and $$a_2$$ lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?

Let S be the set of integers x such that:

1) 100 <= x <= 200

2) x is odd

3) x is divisible by 3 but not by 7.

How many elements does S contain?

Let S be the set of prime numbers greater than or equal to 2 and less than 100. Multiply all the elements of S. With how many consecutive zeroes will the product end?

Let x, y and z be distinct integers, that are odd and positive. Which one of the following statements cannot be true?

Let N = 1421 * 1423 * 1425. What is the remainder when N is divided by 12?

The integers 34041 and 32506 when divided by a three-digit integer n leave the same remainder. What is n?

Each of the numbers $$x_1, x_2, ... ,x_n$$ $$(n > 4)$$, is equal to 1 or -1.
Suppose, $$x_1x_2x_3x_4 + x_2x_3x_4x_5 + x_3x_4x_5x_6 + ....... + x_{n-3}x_{n-2} x_{n-1}x_n x_1 + x_{n-1} x_n x_1 x_2 + x_n x_1 x_2x_3$$ = 0, then:

Let N = $$55^3 + 17^3 - 72^3$$. N is divisible by:

If $$x^2 + y^2 = 0.1$$ and |x-y|=0.2, then |x|+|y| is equal to:

Convert the number 1982 from base 10 to base 12. The result is:

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

A. n is odd

B. n is prime

C. n is a perfect square

If $$n^2 = 123456787654321$$, what is $$n$$?

Let a, b, c be distinct digits. Consider a two digit number $$'ab'$$ and a three digit number $$'ccb'$$, both defined under the usual decimal number system. If ($$ab^{2} = ccb$$) and $$ccb > 300$$ then the value of b is

The remainder when $$7^{84}$$ is divided by $$342$$ is :

$$n^3$$ is odd. Which of the following statement(s) is/are true?
I. $$n$$ is odd.
II.$$n^2$$ is odd.
III.$$n^2$$ is even.

$$(BE)^2 = MPB$$, where B, E, M and P are distinct integers. Then M =

Five-digit numbers are formed using only 0, 1, 2, 3, 4 exactly once. What is the difference between the maximum and minimum number that can be formed?

How many five digit numbers can be formed from 1, 2, 3, 4, 5, without repetition, when the digit at the unit’s place must be greater than that in the ten’s place?

A certain number, when divided by 899, leaves a remainder 63. Find the remainder when the same number is divided by 29.

A is the set of positive integers such that when divided by 2, 3, 4, 5, 6 leaves the remainders 1, 2, 3, 4, 5 respectively. How many integers between 0 and 100 belong to set A?

Number of students who have opted for subjects A, B and C are 60, 84 and 108 respectively. The examination is to be conducted for these students such that only the students of the same subject are allowed in one room. Also the number of students in each room must be same. What is the minimum number of rooms that should be arranged to meet all these conditions?

How many five-digit numbers can be formed using the digits 2, 3, 8, 7, 5 exactly once such that the number is divisible by 125?

What is the digit in the unit’s place of $$2^{51}$$?

A number is formed by writing first 54 natural numbers next to each other as 12345678910111213 ... Find the remainder when this number is divided by 8.

If n is an integer, how many values of n will give an integral value of $$\frac{(16n^2+ 7n+6)}{n}$$ ?

A student instead of finding the value of 7/8 of a number, found the value of 7/18 of the number. If his answer differed from the actual one by 770, find the number.

P and Q are two positive integers such that PQ = 64. Which of the following cannot be the value of P + Q?

If m and n are integers divisible by 5, which of the following is not necessarily true?

P, Q and R are three consecutive odd numbers in ascending order. If the value of three times P is 3 less than two times R, find the value of R.

ABC is a three-digit number in which A > 0. The value of ABC is equal to the sum of the factorials of its three digits. What is the value of B?

A, B and C are defined as follows.

A=$$( 2.000004) \div ((2.000004)^2+ 4.000008)$$ ;

B = $$(3.000003) \div ((3.000003)^2+9.000009)$$

C= $$(4.000002) \div ((4.000002)^2 + 8.000004)$$

Which of the following is true about the values of the above three expressions?

If n is any odd number greater than 1, then $$n(n^2 - 1)$$ is

If a number 774958A96B is to be divisible by 8 and 9, the respective values of A and B will be

How many 3 - digit even number can you form such that if one of the digits is 5, the following digit must be 7?

Three times the first of three consecutive odd integers is 3 more than twice the third. What is the third integer?

The sum of two integers is 10 and the sum of their reciprocals is 5/12. Then the larger of these integers is

What is the greatest power of 5 which can divide 80! exactly?

A third standard teacher gave a simple multiplication exercise to the kids. But one kid reversed the digits of both the numbers and carried out the multiplication and found that the product was exactly the same as the one expected by the teacher. Only one of the following pairs of numbers will fit in the description of the exercise. Which one is that?

Find the minimum integral value of n such that the division $$\frac{55n}{124}$$ leaves no remainder.

Let k be a positive integer such that k+4 is divisible by 7. Then the smallest positive integer n, greater than 2, such that k+2n is divisible by 7 equals

In Sivakasi, each boy's quota of match sticks to fill into boxes is not more than 200 per session. If he reduces the number of sticks per box by 25, he can fill 3 more boxes with the total number of sticks assigned to him. Which of the following is the possible number of sticks assigned to each boy?

If x is a positive integer such that 2x +12 is perfectly divisible by x, then the number of possible values of x is

A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let $$p$$ be a prime number greater than 5. Then $$(p^2-1)$$ is

To decide whether a number of n digits is divisible by 7, we can define a process by which its magnitude is reduced as follows: $$(i_{1}, i_{2}, i_{3}$$,..... are the digits of the number, starting from the most significant digit). $$i_{1} i_{2} ... i_{n} => i_{1}.3^{n-1} + i_{2}.3^{n-2} + ... + i_{n}.3^0$$.
e.g. $$259 => 2.3^2 + 5.3^1 + 9.3^0 = 18 + 15 + 9 = 42$$
Ultimately the resulting number will be seven after repeating the above process a certain number of times. After how many such stages, does the number 203 reduce to 7?

If 8 + 12 = 2, 7 + 14 = 3 then 10 + 18 = ?

The remainder when $$2^{60}$$ is divided by 5 equals

Mr.X enters a positive integer Y(>1) in an electronic calculator and then goes on pressing the square . root key repeatedly. Then

Let a, b be any positive integers and x = 0 or 1, then

If n is any positive integer, then $$n^{3} - n$$ is divisible