Course sections

1
The smallest number which when divided by 4, 6 or 7 leaves a remainder of 2, is (1993)
2
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]
3
The remainder obtained when a prime number greater than 6 is divided by 6 is [1995]
4
5^6-1 is divisible by a. 5 b.13 c. 31 d. None of these. breaking a larger exponent (1995)
5
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]
6
Three consecutive positive even numbers are such that thrice the first number exceeds double the third by 2, then the third number is   [1995].
7
If a number 774958A96B is to be divisible by 8 and 9 (1996)
8
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]
9
Consider a sequence, where the nth term is tn = n/n 2. The value of t3 * t4 * t5 * … t53 equals(2000)
10
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)
11
7^6n- 6^6n, where n is an integer >0, is divisible by (2002)
12
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)
13
What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7 ? ( 2003)
14
What is the remainder when 4^96 is divided by 6 ? (2003)
15
Which among 2^1/2, 3^1/3, 4^1/4, 12^/12 is the largest? (2004)
16
The remainder, when 15^23 23^23 is divided by 19 is? a.4        b.15        c.0        d.18 (2004)
17
Let S be a set of positive integers such that every element n of S satisfies the conditions (2005)
18
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)
19
a/b=1/3, b/c=2, c/d=1/2, d/e=3 then what is the value of abc/def? (2006)
20
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)
21
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)
22
What is the number of distinct terms in the expansion of (a b c)^20? (2008)
23
What is the remainder when 7^74 
24
In a tournament, there are n teams T1, T2 ….., Tn with n > 5. Each team consists of k players, k > 3… (2007)
25
a^2-ab b^2/a^2 ab b^2=1/3 then find a/b (CAT, 2009)
26
Let Sn denote the sum of squares of the first n odd natural numbers. If Sn = 533n, find the value of n. (2011)
27
The value of numbers  2^2004  and 5^2004 are written one after another. How many digits are there in all? (2011)
28
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)
29
What is the sum of all two digit numbers which leave a remainder of 6 when divided by 8? (CAT, 2012)
30
Find the remainder of 21040 divided by 131. (CAT, 2012)
31
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)
32
If a1=1 and a(n 1)-3an 2=4n for every positive integer n, then a100 equals ? (2014)
33
Suppose n is an integer, such that the sum of the digits of n is 2, and 10^10<N<10^11
34
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)
35
What is the greatest power of 5 that divides 80! exactly (2016)
36
If 92x-1- 81x-1= 1944, then x is (2017)
37
The right most non-zero digit of 30^2720 is? (2005)
38
Each family in a locality has at most two adults, and no family has fewer than three children. .. (2004)
39
Let x and y be positive integers such that x is prime and y is composite. (2003)
40
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)
41
The number of common terms in the two sequences 17, 21, 25, …, 417 and 16, 21, 26, …, 466 is (2008)
42
What is the remainder when 7 ^ 74 – 5 ^74 is divided by 4? (2008)
43
if a^2-ab b^2/a^2 ab b^2=1/3, then find (2009)
44
The largest number among the following that will perfectly… (2010)
45
How many integers, greater than 999 but not greater than 4000…(2010)
46
Find the smallest number which when increased by 5 is exactly divisible by 8, 11 and 24. (1994)
47
A student instead of finding the value of 7/8 of a number, found the value of 7/18 of the number (1997)
48
If n^3 is odd, which of the following statements are true? (1998)
49
What is the digit in unit’s place of 2^51? (1998)
50
If n=1+x, where x is the product of four consecutive positive integers, then which of the following is/are true? (1999)
51
The integers 34041 and 32506, when divided by a three digit integer n, leave the same remainder. What is the value of n? ( 2000)
52
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 number of common terms in the two sequences 17, 21, 25, …, 417 and 16, 21, 26, …, 466 is (2008)

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The number of common terms in the two sequences 17, 21, 25, …, 417 and 16, 21, 26, …, 466 is 

a. 19         b.20         c.77          d.78   [CAT, 2008]

  

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Solution: 

 

Arithmetic Sequence /Arithmetic Progression (A.P)

 
First, understand that a sequence is a list of numbers that have a particular pattern. 
A sequence is also called arithmetic sequence or arithmetic progression.
In the question, If the sum of the first 11 terms of arithmetic progression, it was told that what is a sequence and how to find the nth term and the sum of all the terms of a sequence. 
Here, we have been asked a different thing. So. let’s solve this problem with better understanding.
 
The question is saying that find the number of common terms in the two sequences, this means we have to first find the number of terms ( how many numbers are there in each sequence having a common difference) in each of the sequences. Only then the number of common terms in both of the sequences can be found.
The formula to find the number of terms in a sequence is : 
  
{ a }_{ n }={ a }_{ 1 }+(n-1)d   
 where  { a }_{ 1 } is the first term of the sequence and n is the number of the term we want to find out and { a }_{ n }  is term find using n on the RHS if we put the last term  on the LHS i.e.  for { a }_{ n }; on solving for the n on the RHS, we will get the number of terms in the sequence.  And, this is what we need to solve the first part of the question. d= the common difference between two consecutive terms= the next term- the previous term.
So, let’s put the values into the formula. {a}_{1}=17= the first term of the sequence.
{a}_{n}=417= the last term of the sequence.
 n= the number of terms in the sequence given.
d= the next term- the previous term. 21-17=4, 25-21=4
{417} ={ 17}+(n-1)4
{ 417 }={ 17}+4n-4
{ 417 }=13+4n
{ 417 -13}=4n
{ 404}=4n
\frac { 404 }{ 4 } =101
So, the number of terms in the first sequence is n=101. Now, we will find the number of the number of terms in the second sequence using the same formula and then we will find the common terms.
So, let’s put the values into the formula. {a}_{1}=16= the first term of the sequence.
{a}_{n}=466= the last term of the sequence.
p= the number of terms in the sequence given.
d= the next term- the previous term. 21-16=5, 26-21=5
{466} ={ 16}+(p-1)5
{ 466 }={ 16}+5p-5
{ 466}=11+5p
{ 466 -11}=5p
{ 455}=5p
\frac { 455 }{ 5 } =91
So, the number of terms in the second sequence is p= 91.
Now, to find the number of common terms, we have to equate that expression from each sequence through which we found the number of terms in each sequence.
 In the first sequence, we had { 417 }=13+4n.
 And in the second sequence, we had { 466 }=11+5p
11+5p=13+4n
5p=13-11+4n
5p-=2+4n
 Now, find the first term and the common difference, d, for finding the number of common terms in both of the sequences. 
2 should be the first term in the list of common terms because it is just the difference between two first terms 13 and 11.
5p-4n=13-11,                                  13-11=2
The other numbers will appear with a common difference, d, of 5
Why 5 because the number 5 is a divisor in the equation p=\frac { 2+4n }{ 5 }  and division is just another name of subtraction.
Ex. \frac { 10}{ 5 }=2,
This means subtract 5 from 10 twice, 10-5-5=0, and the dividend will be totally divided.
Now the resulting sequence with these numbers starting with 2 with a common difference 5 is:
2, 7,12…97.     
It is ending at 97 because after this we get 102=97+5 which is greater than the numbers of terms in the first sequence.
Now we can find out the number of terms common in both the sequences by finding out the number of terms in the above resulting sequence.
97=2+(c-1)5
97=2+5c-5
97=2-5-5c
97=-3-5c
97+3=5c
c=20=\frac { 100 }{ 5 }
Hence the option (b) is correct and the answer is 20. 

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