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

If n=1+x, where x is the product of four consecutive positive integers, then which of the following is/are true? (1999)

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If n=1+x, where x is the product of four consecutive positive integers, then which of the following is/are true? [CAT, 1999]

     A. N is odd          B. n is prime    C. n is a perfect square       

     a. A and C only    b. A and B only          c. A only         d. None of these

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Solution:

The product of four consecutive positive integers:

 

Here, four consecutive numbers mean if the first number is chosen at random, then the other three numbers which come one after another successively are chosen automatically.

 

That is,

 

(1) If 1 is chosen as the first number, then the other three consecutive numbers are 2, 3, and 4.

 

 

If the number 10 is chosen as the first number, then the other three consecutive numbers are 11, 12, and 13.

 

 

 

The product of numbers means they are being multiplied. AxBxCxD= DxCxBxA = C; C is the product/result of this multiplication.

Interchanging the place of numbers, AxBxCxD= DxCxBxA does not change the product, C, because the multiplication of whole numbers possesses the commutative property.

 

Positive Integers are 1, 2, 3, 4, 5…Z+

 

 

Odd numbers are not divisible by 2.

 

Ex

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 etc

 

 

Prime numbers are either divisible by 1 or by the number itself.

 

Prime numbers from 1 to 100:

 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

 

 

 

Every prime number, leaving 2 alone, is an odd number, but every odd number is not a prime number. 2 is the only prime number which is even. 9, 15, 21 etc. are odd numbers, but they are not prime numbers.

 

 

The number 1 is an odd number. Negative numbers are not prime numbers. The number 15 is an odd number, but it is not a prime number because it is divisible by 3 and 5 also.

 

 

 

When a number n is multiplied by n (by the number itself) its product becomes a perfect square.

 

     n x n =n2

 

Ex. 2×2 =4;     n=2

 

 

3×3= 9;            n=3

 

 

4×4= 12;          n=4

 

 

Here the product of the four consecutive positive integers is designated x. To which, 1 must be added to find N.

 

All three conditions will be checked for N, not for x which is the product of any four consecutive positive integers.

 

 

The best decision criterion in such problem is to prove that the product of any of the four consecutive integers is a prime number by taking four small numbers such as 1x2x3x4= 25. Once you prove that it is a prime number, this means it is also an odd number, but the opposite is not true.

 

 

For example:

1x2x3x4= x=24, x+1=25      it is an odd number and a perfect square of 5 but not a prime number

 

 

Here, we should stop checking the given conditions because at least in one case, we have found the second condition is wrong. That is sufficient to disapprove the second condition, and conclude that only the conditions A and C are correct; no matter in other cases the condition B turns out to be correct. 

(Because the question is saying for all n=x+1)

 

 

 

However, to understand this type of question perfectly, we are expanding the solution.

 

 

 

2x3x4x5= x=120, x+1=121; it is an odd number and a perfect square of 11. But it is also a prime number

 

 

3x4x5x6=x=360. x+1= 361 it is an odd number and a perfect square of 11.

But it is also a prime number.

 

 

Checking Condition—- B. n is prime or not

 

 

To identify a prime number when N is greater than 100,

 

 

 

 Firstly find out the square root of that number (N), and

 

 

 

Then, find the prime numbers coming before the square root of that number.

 

 

 

 If the given number, N, is divisible by any of the prime numbers coming before its square root, then it (N) is not a prime number and if not divisible; then it is a prime number.

 

 

 

The square root of 361 is 19 and the prime numbers coming before it are:

 

 

2, 3, 5, 7, 11, 13, 17

 

 

Since 361(N) is not divisible by any of the prime numbers listed above. It is a prime number.

 

 

Mostly, a prime number never has a unit digit of 0, 2, 4, 5, 6 or 8 and the sum of the digits of that number never becomes the multiple of 3.

 

 

Hence, all three conditions are true in this case.

 

 

For example if we choose 10x11x12x13= 17,160

 

 

This means x=17160, and add 1 to it

 

 

X+1=17161

 

It is a perfect square of 131.

 

 

The sum of the digit is 1+7+1+6+1=16 which is not a multiple of 3. Hence it is a prime number, and all the three conditions are true.

 

 

In all the four examples above:

 

At least N was always an odd number.

 

At least N was always a perfect square.

 

But it was not a prime number in case of the first four positive integers i.e. 1x2x3x4, but it turned out to be a prime number in case of four positive consecutive integers starting from 2x3x4x5.

 

 

Only the conditions (A) and (C) are found to be correct in all cases but not the condition (B).

 

Hence the option (a) is correct.

 

 

  

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