HCF and LCM
Important points to remember :
1. HCF of two or more than two numbers is the greatest number that
divides each of them exactly. It is also known as Greatest Common
Measure or Greatest Common Divisor.
2. LCM (Least Common Multiple) : The least number which is exactly
divisible by each one of the given numbers is called their LCM.
3. The product of two numbers is equal to the product of their
HCF and LCM.
4. HCF of any consecutive numbers is 1.
5. HCF and LCM of fractions :
(i) HCF = HCF of Numerators divided by
LCM of Denominators
(ii) LCM = LCM of Numerators divided by
HCF of denominators
Model Questions:
(1) Two numbers are in the ratio of 15 :11. If their HCF is 13,
find the numbers.
[Solution: Let the numbers be 15 x and 11 x, and so x = 13 (HCF)
So the numbers are 15 x 13, and 11 x 13
Ans : 195 and 143.]
(2) Two numbers are in the ratio of 3 : 4, and their HCF is 4.
Find their LCM.
[Hint : Two numbers are 3 x 4(HCF) and 4 x 4 (HCF),
and so their LCM is 3 x 4 x 4 = 48 (Ans)]
(3) Two numbers are in the ratio of 2 : 3. If their LCM is 48,
find the numbers.
(4) The HCF and LCM of two numbers are 11, and 693 respectively.
If one number is 77, find the other.
(5) The product of two numbers is 1320. If their HCF is 6,
find their LCM.
(6) The sum of two numbers is 216. If their HCF is 27, find the numbers.
(7) The sum of two numbers is 2000. If their LCM is 21879,
find the numbers.
(8) The LCM and HCF of two numbers are 495 and 5 respectively.
If their sum is10, find the numbers.
(9) The product of LCM and HCF of two numbers is 24. If their
difference is 2, find the numbers.
(10) The HCF and LCM of two numbers are 84 and 21 respectively.
If the numbers are in the ratio of 1 : 4, find the larger number.
(11) Find the greatest possible length which can measure exact;y
the length of 4m 95cm, 9m16cm, and 16m65cm.
(12) The length , breadth, and height of a room are 8m25cm, 6m75cm,
and 4m50cm respectively. Determine the longest rope which can
measure the three dimensions of the room exactly.
(13 Two tankers contain 850 lt and 680 lt of petrol respectively. Find
the maximum capacity of a container, which can measure the
petrol of either tank in exact number of times.
(14) A rectangular courtyard is 20m16cm long and 15m60cm broad.
It is to be paved with square stones of same size. Find the least
possible number of such tones.
[Hint: HCF of 2016 and 1560 is 24,
No. of square stones required = length x breadth
divided by HCF x HCF
= 65 x 84
= 5460 (Ans)]
(15) Find a rod of the greatest length which can measure exactly
42m, 49m, 63m.
(16) Two bills of Rs. 6075 and Rs.8505 respectively are to be paid
by cheques of the same amount. What will be the largest
possible amount of each cheque ?
(17) There are four bells, which toll at intervels of 3,7,12, and 14
seconds respectively. The four bells begin to toll at 12 noon.
How often will they toll together in 14 minutes ?
[Hint: LCM of 3,7,12,and 14 is 84
No. of times they toll together in 14 minutes, i.e.,
in 14 x 60 seconds = 14 x 60
divided by LCM
i.e., 14 x 60 divided by 84 = 10 times (Ans)]
(18) 4 bells toll after an intervel of 8,9,12, and 15 seconds
respectively. When they toll again ?
(19) The circumference of the wheels of a carriage are 3m25cm,
and 5m. What is the least distance in which both the wheels
make an exact number of revolutions ?
(20) A boy saves Rs 4.65 daily. Find the least no. of days
in which he will be able to save an exact number of rupees.
[Hint: The exact number of rupees will be a multiple of 100.
LCM of 100 and 465 = 5 x 20 x 93 = 9300
i.e., the boy saves 9300 paise
Hence, the no. of days required = 9300 divided by 465
= 20 (Ans)]
(21) Electric posts occur at equal distance of 220m along a road
and police constables are standing at equal distance of 300m
along the same road. The first constable is standing at the foot
of the first electric post. How far from it along the road is the
next constable standing at the foot of an electric post ?
[Hint: The required distance = LCM of 220 and 300]
(22) In a walking competition three persons step off together.
Their steps measure 85cm, 90cm, and 80cm respectively.
At what distance from the starting point will they again
step off together ?
(23) Three different containers contain 496, 403 and 713 litres
of milk respectively. Find the biggest jug which can measure
all the different quantities exactly.
(24) A rectangular courtyard 3.78 mt long and 5.25 mt wide is
to be paved exactly with square tiles, all of the same size.
What is the largest size of the tile which could be used for
the purpose ?
(25) Four differenct electronic devices make a beep after every
30 minutes, 1 hour, 1 1/2 hour and 1 hour 45 minutes
respectively. All the devices beeped together at 12 noon.
(i) At what time will they all beep together again ?
(ii) In 30 minutes, how many times do they beep together ?
(26) A, B, and C start at the same time in the same direction to
run around a circular stadium. A completes a round in 252
seconds, B in 308 seconds, and C in 198 seconds, all starting
at the same point. After what time will they meet again at the
starting point ?
(27) The traffic signal lights at three different road crossings change
after every 48 seconds, 72 seconds, and 108 seconds respectively.
If they all change simultaneously at 08:20 hours, then at what
time will they again change simultaneously ?
[Solution: Interval of change = LCM of 48,72, 108 seconds.
Hence, the lights will again change simultaneously
after every 432 seconds, i.e., 7 minutes 12 seconds.
So, the next simultaneous change will take place at
8:27:12 hrs.]
(28) Find the least number which when divided by 5,6,7, and 8
leaves a remainder of 3, but when divided by 9 leaves no
remainder.
[Solution : LCM of 5,6,7,8 = 840
So, the required number is of the form 840k + 3
Least value of k for which 840k +3 is divisible by 9
is k = 2
Hence the required number is 840 x 2 + 3 = 1683]
(29) The LCM of two numbers is 45 times their HCF. If one of
the number is 125 and the sum of HCF and LCF is 1150,
find the other number.
(30) Find the maximum number of students among whom 1001
pens and 916 pencils can be distributed in a such a way
that each student gets the same number of pens and same
number of pencils.
::: Compiled by N. Ramadhas, Nagercoil-629001
Mobile No: 9486077890
No comments:
Post a Comment