# Question Bank - Number theory - Sibanand Pattnaik

• Q88) If p is any three digit number and q is any number obtained with any type of permutations of the digits of p, then p – q is always divisible by
(a) 2
(b) 3
(c) 6
(d) 9

• Q89) How many two digit or three digit positive integers in base 6 are there such that if 0 in inserted between the units and the tens digit, the multiple of the original number is obtained?

• Q90) If p=abc and Q=uv are three digits and two digits natural numbers respectively, such that u and v must be distinct integers.How many pairs of P and Q are there in total which gives the same result when we multiply abc with uv as the product of cba with vu?
a. 2
b. 8
c. 5
d. cbd

• Q91) Let 123abc231bca312cab be a 18 digit number in base 7. How many ordered pairs (a,b,c) exist such that the given 18 digit number is divisible by 4?

• Q92) Find the sum of all 3-digit positive numbers N that satisfy the condition that the digit sum of N is 3 times the digit sum of N + 3.

• Q93) How many four digit numbers are there in which the square of sum of first two digit & last two digit equals to the number itself ?

• Q94) There are Five consecutive positive integers whose sum is a cube and sum of middle three numbers is a square. Find the sum of the digit of smallest possible middle number ?

• Q95) Consider two operations which can be performed on a fraction:

1. increase the numerator by 8
2. increase the denominator by 7

Having performed a total number of n such operations in some order, starting with the fraction 7/8 we obtain a fraction of equal value. What is the smallest possible value of n?

• Q96) Ioana picks out two numbers a and b from the set {1, 2, 3, …, 26}. The product ab is equal to the sum of the remaining 24 numbers. What is the value of |a−b|?

• Q97) If all the five digit numbers that can be formed using the digits 1, 3, 5, 7 and 9 are arranged in increasing order, the 500th number is ?

• Q98) 100 numbers are written around a circle. The sum of every 8 consecutive numbers is 25/2. The 9th number is -1/2, the 19th number is 3/4 and 20th number is 2. What is the 50th number?
(a) 4
(b) 5
(c) 1
(d) none of the foregoing

• Q99) All numbers from 1 to 200 (in decimal system) are written in base 7 notation. How many of these will not contain any zero?

• Q100) A teacher writes the numbers 1 and 2 on the blackboard. Each student, one at a time, goes to the board, chooses two numbers x and y that are written on the board (possibly x = y), computes x + y + xy, and writes the result on the board. When the students are ﬁnished, there are several numbers on the board, including 1727, 2303, 2467, 3455, and 7775. Unfortunately, one of these numbers was produced by a student using a calculator whose batteries were weak. Which of the ﬁve numbers cannot be produced by correct calculations by the students?
a. 1727
b. 2303
c. 2467
d. 3455
e. 7775

• @sibanand_pattnaik

• @sibanand_pattnaik 1. Even = 45
2. Odd = 15

How to find the 3 one ?

• @sibanand_pattnaik Sum = 152295
Product = (45000)^30

• @sibanand_pattnaik

2^(n-1)

here n = number of primes

so 2^(3-1) = 4

• @sibanand_pattnaik
45000 * 6000

161

54

138

42

45

30

53