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The smallest number which when divided by 4, 6 or 7 leaves a remainder of 2, is (1993)
Which is the least number that must be subtracted from 1856 so that the remainder, when divided by 7, 12, and 16, is 4? [1994]
The remainder obtained when a prime number greater than 6 is divided by 6 is [1995]
5^6-1 is divisible by a. 5 b.13 c. 31 d. None of these. breaking a larger exponent (1995)
Two positive integers differ by 4 and sum of their reciprocals is 10/21, then one of the numbers is a.  3 b.  1 c.  5 d. 21 [1995]
Three consecutive positive even numbers are such that thrice the first number exceeds double the third by 2, then the third number is   [1995].
If a number 774958A96B is to be divisible by 8 and 9 (1996)
If m and n are integers divisible by 5, which of the following is not necessarily true? a.  m – n is divisible by 5 [1997]
Consider a sequence, where the nth term is tn = n/n 2. The value of t3 * t4 * t5 * … t53 equals(2000)
A red light flashes 3 times per minute and a green light flashes 5 times in two minutes at regular intervals. If both lights start flashing at the same time, how many times do they flash together in each hour? ( 2001)
7^6n- 6^6n, where n is an integer >0, is divisible by (2002)
A child was asked to add first few natural numbers (that is, 1 2 3……) so long his patience permitted. As he stopped, he gave the sum as 575. When the teacher declared the result wrong the child discovered he had missed one number in the sequence during addition. The number he missed was: (2002)
What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7 ? ( 2003)
What is the remainder when 4^96 is divided by 6 ? (2003)
Which among 2^1/2, 3^1/3, 4^1/4, 12^/12 is the largest? (2004)
The remainder, when 15^23 23^23 is divided by 19 is? a.4        b.15        c.0        d.18 (2004)
Let S be a set of positive integers such that every element n of S satisfies the conditions (2005)
When you reverse the digits of the number 13, the number increases by 18. How many other 2 digit numbers increase by 18 when their digits are reversed?  ( 2006)
a/b=1/3, b/c=2, c/d=1/2, d/e=3 then what is the value of abc/def? (2006)
Consider four-digit numbers for which the first two digits are equal and the last two digits are also equal. How many are such numbers perfect squares? (2007)
how-many-pairs-of-positive-integers-m-n-satisfy 1/m 4/n=1/12, where n is an odd number less than 60…. (2007)
What is the number of distinct terms in the expansion of (a b c)^20? (2008)
What is the remainder when 7^74 
In a tournament, there are n teams T1, T2 ….., Tn with n > 5. Each team consists of k players, k > 3… (2007)
a^2-ab b^2/a^2 ab b^2=1/3 then find a/b (CAT, 2009)
Let Sn denote the sum of squares of the first n odd natural numbers. If Sn = 533n, find the value of n. (2011)
The value of numbers  2^2004  and 5^2004 are written one after another. How many digits are there in all? (2011)
If p be a prime number, p>3 and let x be the product of positive number 1, 2, 3,…,(p-1), then consider the following statements (2012)
What is the sum of all two digit numbers which leave a remainder of 6 when divided by 8? (CAT, 2012)
Find the remainder of 21040 divided by 131. (CAT, 2012)
A three-digit number which on being subtracted from another three-digit number consisting of the same digits in reverse order gives 594. The minimum possible sum of all the three digits of this number is (2013)
If a1=1 and a(n 1)-3an 2=4n for every positive integer n, then a100 equals ? (2014)
Suppose n is an integer, such that the sum of the digits of n is 2, and 10^10<N<10^11
A sequence of 4 digits, when considered as a number in base 10 is four times the number it represents in base 6. What is the sum of the digits of the sequence? (2016)
What is the greatest power of 5 that divides 80! exactly (2016)
If 92x-1- 81x-1= 1944, then x is (2017)
The right most non-zero digit of 30^2720 is? (2005)
Each family in a locality has at most two adults, and no family has fewer than three children. .. (2004)
Let x and y be positive integers such that x is prime and y is composite. (2003)
If sum of the first 11 terms of arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms? (2004)
The number of common terms in the two sequences 17, 21, 25, …, 417 and 16, 21, 26, …, 466 is (2008)
What is the remainder when 7 ^ 74 – 5 ^74 is divided by 4? (2008)
if a^2-ab b^2/a^2 ab b^2=1/3, then find (2009)
The largest number among the following that will perfectly… (2010)
How many integers, greater than 999 but not greater than 4000…(2010)
Find the smallest number which when increased by 5 is exactly divisible by 8, 11 and 24. (1994)
A student instead of finding the value of 7/8 of a number, found the value of 7/18 of the number (1997)
If n^3 is odd, which of the following statements are true? (1998)
What is the digit in unit’s place of 2^51? (1998)
If n=1+x, where x is the product of four consecutive positive integers, then which of the following is/are true? (1999)
The integers 34041 and 32506, when divided by a three digit integer n, leave the same remainder. What is the value of n? ( 2000)
Anita had to do a multiplication. Instead 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? (2001)

Section 1

The integers 34041 and 32506, when divided by a three digit integer n, leave the same remainder. What is the value of n? ( 2000)

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The integers 34041 and 32506, when divided by a three-digit integer n, leave the same remainder. What is the value of n? [CAT, 2000]

(a) 289             (b) 367             (c) 453             (d) 307 Protection Status





To solve this question, we have to use the Remainder Theorem


Dividend= (Divisor x Quotient) + Remainder





D= dq + r


In  \frac { 11 }{ 2 }

D=11, d=2, q=5, and r=1


11= (2 x 5) + 1


If the value of any one of the three values is not known on the

right-hand side, it can be found by using cross multiplication.


(On the left-hand side, it is assumed that we have at least one value as in our case)



D=11, d=?, q=5, r=1



11= (dx 5) + 1






\frac { 10 }{ 5 } = 2 = d



If the value of any two of the three values is not known on the


right-hand side, the equation cannot be found by using cross

multiplication until another equation is brought and a relation

between them is established in such a way that we can

approximate the value of one of the two unknowns.




D=11, d=?, q=? r=1

D=21 d=?, q=? r=1



By the concept of the same remainder, a relationship can be established between them as


(21-1) – (11-1) =10



Since it is known that if the two dividends are divisible by the same divisor, their difference will also be divisible by the same divisor.


Ex. \frac { 15 }{ 3 };    \frac { 24 }{ 3 }


24-15 = 9 which is divisible by 3


Following the above exposition, we can now find a common dividend that can give us a common divisor required to solve both the equations.



As we have found this in our case:


(21-1) – (11-1) =10


10 is the common dividend which will give us the common divisor required to solve both equations.



Take factors of 10 (factors mean divisors of 10 which always range from 1 to   \frac { n }{ 2 } if n is the number for which factors are being found, n is being ignored as a factor)


1, 2, 5, (10)



Now we have found three possible values of our divisor (ignoring

10 as a factor of 10), we will fit each of these three into any of the

two equations to see which one solves it.


11= (2 x 1) + 1              (can’t find)


11= (2 x 2) + 1            (can’t find)



11=(2×5)+1                 (yes)



So 5 is the divisor which can solve both equations.



In the same way, the given question can be solved:


For 34041


34041= (Divisor x Quotient) + Remainder



Nothing is known on the right-hand side



32506= (Divisor x Quotient) + Remainder


But, the question also gives a common relation between these two equations by stating “when divided by a three digit integer n, leaves the same remainder.”



This means the remainder is the same as it was 1 in our example given above.



We will follow the same procedure as shown above.


34041= (Divisor x Quotient) + R

32506= (Divisor x Quotient) + R                        Or


34041-R= (Divisor x Quotient)       

32506-R= (Divisor x Quotient)



If the two dividends are divisible by the same divisor, their difference will also be divisible by the same divisor. (Shown above)



(34041-R)-( 32506-R)




34041-R-32506+R   (bracket is opened)



34041 -32506+R –R                                       +R –R= cancelled



34041 -32506= 1535



Now find the factors of 1535, which are:

1, 5, 307, (1535)






 Find from among the options that which one divides 1535 completely. (In competitive exams (MCQ type)), this approach is desired because you have to choose any of the given options as your answer).



Our factor 307 matches with the option (d)

 (a) 289            (b)367              (c) 453             (d)307


34041= (307 x Quotient) + R

32506= (307 x Quotient) + R






34041= (307 x 110) + 271

32506= (307 x 105) + 271


Hence the option (d) is correct.


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