Session Numbers

LCM & HCF – II

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:

• HCF of two or more numbers is the greatest number which completely divides the numbers. In otherwords, it is the greatest factor common to the numbers.

• The largest number N, which divides the numbers n1, n2 and n3 (n1 > n2 > n3) and leaves a remainder

of ‘r’ in each case is given by { }1 2 2 3N = HCF (n n ), (n n )− − .

• The largest number N, which divides the numbers n1, n2 and n3 and leaves remainders of r1, r2 and r3

respectively is given by { }1 1 2 2 3 3N = HCF (n r ),(n r ),(n r )− − − .

� If there are three types of items such that number of items of first type is n1, that of second type is

n2 and that of third type is n

3, then the minimum number of groups (N) in which these items can be

slotted such that all the groups have same number of items and each of the groups has only one

type of item 31 2

1 2 3 1 2 3 1 2 3

nn nN

HCF (n , n , n ) HCF (n , n , n ) HCF (n , n , n )= + + 1 2 3

1 2 3

n n n

HCF (n , n , n )

+ += .

• The LCM of two or more numbers is the least number which is divisible by all the numbers.

• If p q rN a b c ....= × × × is a natural number such that N = LCM (x, y), then the number of distinct

pairs (P) of x and y is given by (2p 1)(2q 1)(2r 1)... 1

P2

+ + + += .

• LCM and HCF of fractions

a c e LCM (a, c, e)LCM , ,

b d f HCF (b, d, f ) =

;

a c e HCF (a, c, e)HCF , ,

b d f LCM (b, d, f ) =

, where a c e

, andb d f

are the fractions in simplest form.

� LCM (N1, N

2) × HCF (N

1, N

2) = N

1 × N

2

The above relation is true for more than two numbers if and only if each of numbers is co-prime torest of the numbers i.e. HCF of any two numbers is 1. Note that, if the HCF of a set of numbers is

equal to 1, the above relation is not necessarily valid.

• 1 2 1 2HCF N + N , LCM (N , N ) = HCF (N1, N2)

SessionNumbers

• The number N, which when divided by n1, n2 and n3 leaves the same remainder of ‘r’ in each case, is

given by 1 2 3N k LCM(n , n , n ) r= × + , where k is a whole number.

• The number N, which when divided by n1, n2 and n3 leaves remainders of r1, r2 and r3 respectively such

that n1 – r1 = n2 – r2 = n3 – r3 = d, is given by 1 2 3N k LCM(n , n , n ) d,= × − where k is a natural number.

• The number N, which when divided n1 and n2 leaves remainders of r1 and r2 such that 1 2r r≠ and

1 1 2 2n – r n – r ,≠ is given by 1 2 LN k LCM(n , n ) N ,= × + where k is a whole number and NL is the

smallest number which satisfies the aforementioned conditions.� Three bells B1, B2 and B3 chime at regular intervals of T1, T2 and T3 min respectively. These bells

chime for c1, c2 and c3 min respectively. If all the bells start chiming together, then the time interval(I) between two successive occasions when they start chiming together again is given byI = LCM (T1 + c1, T2 + c2, T3 + c3) min. If the bells chime for negligible amounts of time, thenI = LCM (T1, T2, T3) min.

Highlight: This session deals with application based questions of LCM & HCF which are of moderatedifficulty level.

Session Numbers

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

Q1. Number of students who have opted for subjects A, B and C are 60, 84 and 108 respectively. Theexamination is to be conducted for these students such that only the students of the same subjectare allowed in one room. Also the number of students in each room must be same. What is theminimum number of rooms that should be arranged to meet all these conditions?(a) 28 (b) 60 (c) 12 (d) 21 (CAT 1998)

Q2. A red light flashes three times per minute and a green light flashes five times in 2 min at regularintervals. If both lights start flashing at the same time, how many times do they flash together ineach hour?(a) 30 (b) 24 (c) 20 (d) 60 (CAT 2001)

DIRECTION for Question 3: 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.

Q3. What is the number x?I. The LCM of x and 18 is 36.II. The HCF of x and 18 is 2. (CAT 1995)

Q4. 21 mango trees, 42 apple trees and 56 orange trees have to be planted in rows such that each rowcontains the same number of trees of one variety only. Minimum number of rows in which the treesmay be planted is:(a) 20 (b) 17 (c) 15 (d) 4 (FMS 2005)

Q5. Three Vice Presidents (VP) regularly visit the plant on different days. Due to labour unrest, VP (HR)regularly visits the plant after a gap of 2 days. VP (Operations) regularly visits the plant after a gapof 3 days. VP (Sales) regularly visits the plant after a gap of 5 days. The VPs do not deviate fromtheir individual schedules. CEO of the company meets the VPs when all the three VPs come to theplant together. CEO is on leave from January 5th to January 28th, 2012. Last time CEO met the VPson January 3, 2012. When is the next time CEO will meet all the VPs?(a) February 6, 2012 (b) February 7, 2012 (c) February 8, 2012(d) February 9, 2012 (e) None of the above (XAT 2012)

Q6. At a bookstore, ‘MODERN BOOK STORE’ is flashed using neon lights. The words are individually

flashed at the intervals of 5 17 41

s, s and s2 4 8

respectively, and each word is put off after a second.

The least time after which the full name of the bookstore can be read again is(a) 49.5 s (b) 73.5 s (c) 1744.5 s (d) 855 s (CAT 2002)