Session Numbers

Odd, Even and Prime – I

Numbers is one of the most important topics for CAT and other management entrance exams, questionsfrom which have appeared consistently and in significant numbers in all these exams.

Key concepts discussed:

• All the natural numbers which are multiples of 2 are even numbers. Even numbers are resented as2n, where n is a natural number.

• All the natural numbers which are not multiple of 2 are odd numbers. Odd numbers are representedas 2n – 1, where n is a natural number.

• Facts about even and odd numbers:odd ± odd = evenodd ± even = oddeven ± even = evenodd × odd = oddodd × even = eveneven × even = even

• All the natural numbers, which have exactly two distinct factors i.e. 1 and number itself, are primenumbers.

• There are 25 prime numbers from 1 to 100 and 21 from 101 to 200.• 2 is the only prime number which is even.• Every prime number greater than 3 can be represented as either 6n – 1 or 6n + 1, where n is a

natural number, but the converse is not true (i.e. every number of the above mentioned forms is nota prime number.)

• Prime numbers are the building blocks of composite numbers i.e. every composite number is aproduct of two or more identical or distinct prime numbers.

• A natural number is a prime if it is not divisible by any prime number which is less than or equal tothe square root of the number.

Highlight: Though the session deals with questions which are based on the definitions of numbers, a fewquestions are tricky in nature. The session demonstrates that just mere understanding of definitions wouldnot be adequate to handle tricky questions unless other concepts pertinent to the topic are applied adeptly.

SessionNumbers

The questions discussed in the session are given below along with their source.

Q1. Let S be the set of integers x such that

I. 100 x 200,≤ ≤II. x is odd andIII. x is divisible by 3 but not by 7.How many elements does S contain?(a) 16 (b) 12 (c) 11 (d) 13 (CAT 2000)

DIRECTIONS for Question 2: The question is followed by two statements, I and II.Mark the answer as:(a) if the question can be answered with the help of statement I alone,(b) if the question can be answered with the help of statement II, alone,(c) if both, statement I and statement II are needed to answer the question, and(d) if the question cannot be answered even with the help of both the statements.

Q2. What are the ages of the three brothers?I. The product of their ages is 21.II. The sum of their ages is not divisible by 3. (CAT 1993)

Q3. Let x, y and z be distinct positive integers satisfying x < y < z and x + y + z = k. What is thesmallest value of k that does not determine x, y, z uniquely?(a) 9 (b) 6 (c) 7 (d) 8 (CAT 1993)

Q4. Given odd positive integers x, y and z, which of the following is not necessarily true?(a) x2 y2 z2 is odd (b) 3(x2 + y3)z2 is even (c) 5x + y + z4 is odd (d) z2 (x4 + y4)/2 is even

(CAT 1993)

DIRECTIONS for Questions 5: The question is followed by two statements, I and II.Mark the answer as:(a) if the question can be answered with the help of statement I alone.(b) if the question can be answered with the help of statement II alone.(c) if both statement I and statement II are needed to answer the question.(d) if the question cannot be answered even with the help of both the statements.

Q5. If x, y and z are real numbers, is z – x even or odd?I. xyz is odd.II. xy + yz + zx is even. (CAT 1995)

Q6. The smallest positive value of x for which the fractions

x 2 x 13 x 26 x 41 x 1913 x 2002, , , , , ,

10 11 12 13 49 50+ + + + + +− − −− are in their simplest form is

(a) 47 (b) 49 (c) 51 (d) 53 (JMET 2007)