CAT Algebra

Reading Test 1

Question No : 1

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

Question No : 2

Q.2 Let n and m be two positive integers such that m < n, and there are exactly 41 integers greater than 8^m and less than 8ⁿ, which can be expressed as powers of 2. Then, the smallest possible value of n + m is

Question No : 3

Q.3 For some real numbers a and b, the system of equations x + y = 4 and (a + 5)x + (b² -15)y = 8b has infinite many solutions for x and y. Then, the maximum possible value of ab is

Question No : 4

Q.4 Let n be any natural number such that 5^n-1<3ⁿ⁺¹. Then the least integer value of m that satisfies 3ⁿ⁺¹<2^n+mfor each such n, is

Question No : 5

Q.5 Let a, b, m and n be natural numbers such that a > 1 and b > 1. If a^mbⁿ = 144¹⁴⁵, then the largest possible value of n - m is:

Question No : 6

Q.6 Let n be the least positive integer such that 168 is a factor of 1134ⁿ If m is the least positive integer such that 1134ⁿis a factor of 168^m, then m+n equals

Question No : 7

Q.7 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 No : 8

Q.8 The number of positive integers less than 50, having exactly two distinct factors other than 1 and itself, is:

Question No : 9

Q.9 The sum of the first two natural numbers, each having 15 factors (including 1 and the number itself), is

Question No : 10

Q.10 The number of all natural numbers up to 1000 with non-repeating digits is

Question No : 11

Q.11 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 No : 12

Q.12 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 No : 13

Q.13 For some natural number n, assume that (15,000)! is divisible by (n!)!. The largest possible value of n is

Question No : 14

Q.14 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

Question No : 15

Q.15 For all possible intergers n satisfying 2.25 ≤ 2 + 2ⁿ⁺² ≤ 202, then the number of integer values of 3 + 3^{n + 1} is

Question No : 16

Q.16 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