Let N be a positive integer not equal to 1 . Then, none of the numbers 2 , 3 , … . , N is a divisor of ( N ! - 1 ) . Thus, we can conclude that:

The smallest number between N and N ! , which is a divisor of ( N ! + 1 ) , is a prime number

At least, one of the numbers ( N + 1 ) , ( N + 2 ) , … , ( N ! - 2 ) is a divisor of ( N ! - 1 )

None of the foregoing statements is necessarily correct

E M B I B E

Quantitative Aptitude for the CAT>Chapter 1 - Numbers and Their Properties>Practice Exercises>Q 55

EASY

CAT

IMPORTANT

Earn 100

How many different four-digit numbers are there in the octal (base $8$ ) system when expressed in that system?

50% studentsanswered this correctly

Important Questions on Numbers and Their Properties

EASY

CAT

IMPORTANT

Quantitative Aptitude for the CAT>Chapter 1 - Numbers and Their Properties>Practice Exercises>Q 56

How many numbers less than or equal to

500

are there, each of which is the product of more than three distinct prime numbers?

EASY

CAT

IMPORTANT

Quantitative Aptitude for the CAT>Chapter 1 - Numbers and Their Properties>Practice Exercises>Q 57

How many numbers below

100

can be expressed as a difference of two perfect squares in only one way?

EASY

CAT

IMPORTANT

Quantitative Aptitude for the CAT>Chapter 1 - Numbers and Their Properties>Practice Exercises>Q 58

The LCM of the two quotients obtained when two numbers a and b are divided by their HCF is

77

. LCM of another pair of quotients, when other two numbers c and d have the same HCF as that of a and b, and when divided by it is

221

. Which of the following could be the ratio of the sum of a and b to that of the difference between c and d?

EASY

CAT

IMPORTANT

Quantitative Aptitude for the CAT>Chapter 1 - Numbers and Their Properties>Practice Exercises>Q 59

Let p, q, and s be integers such that

p^{2} = {sq}^{2}

. Then, it follows that:

EASY

CAT

IMPORTANT

Quantitative Aptitude for the CAT>Chapter 1 - Numbers and Their Properties>Practice Exercises>Q 60

Let N be a positive integer not equal to

1

. Then, none of the numbers

2, 3, \dots ., N

is a divisor of

(N! - 1)

. Thus, we can conclude that:

EASY

CAT

IMPORTANT

Quantitative Aptitude for the CAT>Chapter 1 - Numbers and Their Properties>Practice Exercises>Q 1

The product of two positive numbers is

616

. If the ratio of the difference of their cubes to the cube of their difference is

157 : 3

, then the sum of the two numbers is:

EASY

CAT

IMPORTANT

Quantitative Aptitude for the CAT>Chapter 1 - Numbers and Their Properties>Practice Exercises>Q 2

If m and n are integers such that

(\sqrt{2})^{19} \times 3^{4} \times 4^{2} \times 9^{m} \times 8^{n} = 3^{n} \times 16^{m} \times (\sqrt[4]{64})

then m is:

MEDIUM

CAT

IMPORTANT

Quantitative Aptitude for the CAT>Chapter 1 - Numbers and Their Properties>Practice Exercises>Q 3

In a six-digit number, the sixth, that is, the rightmost, digit is the sum of the first three digits, the fifth digit is the sum of first two digits, the third digit is equal to the first digit, the second digit is twice the first digit and the fourth digit is the sum of fifth and sixth digits. Then, the largest possible value of the fourth digit is