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, Lecture 45

How many integers, greater than 999 but not greater than 4000…(2010)

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 How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4 if repetition of digits is allowed? [CAT, 2010]
a. 499     b.  500         c.  375             d.  376

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Solution:

 

 The Fundamental Principle of Counting or the Multiplication Principal :

 
 

 We will use the Fundamental Principle of Counting or the Multiplication Principal to solve this problem.

 

The integer greater than 999 is 1000 and the integer greater than 4000 is 4001.
 
So, we have to find the number of integers from 1000 to 4000.
 
 (all integers will be having four digits because there are four digits in 1000 and in 4000 both)
 
1000 ≤ Z ≥ 4000   where Z is the symbol of Integers.
 
So, between the given interval, there are 3001 integers.
 
The question is actually asking how many integers can be formed having four digits with the following digits 0, 1, 2, 3, and 4.

 

The thousands place, the left-most digit of the numeral cannot have “4” or “o” because:
4001, 4102, 4003 etc., > 4000
Similarly, 0123, 0321 etc., <1000
 
This means we can only have 1, 2 and 3 in the thousands place. We have only three choices for the thousands place.
 
Ex. 3012, 2001, 1022 etc.,
But for the hundreds, tens and units place, we are free to choose any of the given integers i.e., 0, 1, 2, 3 and 4. Since repletion is allowed, any digit from the given digits may appear on the places of remaining digits. Ex, 3432 or 3031 or 3243, etc. All three places, hundreds, tens and units place, can have any of the 5 integers.
 
This means for hundreds, tens and units place, we can choose any of the five integers.
  So, we have five choices in these places.
  

 

So, 3 x 5 x 5 x 5 = 375 and we have to add 1 to it because we didn’t allow the thousands place to hold 4 but we have one numeral which is 4000. So, for 4000 we have to add 1 number more and now it becomes 375+1=376.
Hence the answer is 376 and the option (d) is correct.

 

Further Explanation

 

To better appreciate the above methodology, we can understand the Fundamental Principle of Counting or the Multiplication Principal through the following example:

 

For example, you are to find how many numbers/numerals can be formed using 1, 2, 3 and 4 as digits: (Numeral is just the symbol or numerical representation of a number if you say 1+1+1+1=4. In this case the count is 4, a concept, and the symbol is 4 in written form we can see and both are the same).

 

1. When the repetition of digits is allowed     and
2. When the repetition of digits is not allowed.
When the repetition of digits is allowed:

 

For the first place, all the given four digits, 1, 2, 3 and 4 can occupy the left-most place/the thousands place.  Since there are 4 digits, so N=4 for the first place.

Since repetition of digits is allowed, the second place can also be occupied by all the given four digits, 1, 2, 3 and 4.

 

Again, since the repetition of digits is allowed,  the third and fourth places can be occupied by all the given four digits. This means the digits 1, 2, 3, and 4 can come in the third place like 1211, 1222, 1233, 1244, etc.

 

And also in the fourth place:
1211, 1222, 1233, 1244 etc.
 

 

This is why the following picture emerges.
Now we have N x N x N x N = the numbers/numerals formed with 1, 2, 3 and 4 as digits with repetition.
Where N=4,
So the answer is 4 x 4 x 4 x 4 = 44  = 256 numbers.
When the repetition of digits is not allowed: (N= the Count of the numbers= 4)

 

In this case, after each assignment, we subtract 1 from the N because we cannot use/repeat any assigned/allocated digit.

 

 

For the first place, any of the four digits can occupy this place. This is why we have:

 

Now we have lost one digit from the group of given 1, 2, 3 and 4 and we can, now, only use any three of the remaining digits. Similarly, for the third place, we have any of the two remaining digits and the last place has only one digit.

 

So, 4 x 3 x 2 x 1 = 24 numbers/numerals can be formed using these four digits, 1, 2, 3 and 4 when the repetition of the numbers is not allowed.

 

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