Q 65. LCM AND HCF
What is the smallest number greater than 1000 that when divided by any one of the numbers 6, 9, 12, 15, 18 leaves a remainder of 3?
- LCM of 6, 9, 12, 15, 18 = 180.
- Means this is the smallest number that will be divided by every number.
- Such number above 1000 will also be a multiplication of 180.
- 180 × 5 = 900 < 1000
- 180 × 6 = 1080. > 1000 however this is the smallest multiple of 180 above 1000.
- Now, we need the remainder as 3. So, desired number = 1083.
- This question can also be solved backward by trying every options.
Q 12. LCM AND HCF
If you have two straight sticks of length 7.5 feet and 3.25 feet, what is the minimum length can you measure?
Length of sticks are 7.5 feet ad 3.25 feet
Formula used:
HCF of (A/B) and (C/D) = (HCF of A and C)/(LCM of B and D)
Calculation:
First convert decimal into fraction
⇒ 7.5 = 75/10 = 15/2
⇒ 3.25 = 325/100 = 13/4
Now, minimum length will be measured by HCF of {15/2, 13/4}
⇒ HCF = 1/4 = 0.25
∴ The minimum length can be measured by two stick is 0.25 foot.
Q 51. LCM AND HCF
What is the greatest length x such that 3 1/2 m and 8 3/4 m are integral multiples of x?
3 ½ = 7/2 and 8 ¾= 35/4
HCF (Fractions) = HCF (Numerator) / LCM (Denominator)
x = H. C. F. of (7/2) and (35/4) = H.C.F. of 7 and 35 / L.C.M. of 2 and 4 = 7/4 Hence, x = 7/4 m or 1 3/4 m
Q 54. LCM AND HCF
What is the least four-digit number when divided by 3, 4, 5 and 6 leaves a remainder 2 in each case?
• L.C.M. of 3, 4, 5 and 6 = 60
Let the required number be 60x + 2
If x = 17, then the number = 60 × 17 + 2 = 1020 + 2 = 1022
Q 15. LCM AND HCF
In a school every student is assigned a unique identification number. A student is a football player if and only if the identification number is divisible by 4, whereas a student is a cricketer if and only if the identification number is divisible by 6. If every number from 1 to 100 is assigned to a student, then how many of them play cricket as well as football?
• The required number should be completely divisible by both 4 and 6.
• That means it should be divisible by LCM of 4 and 6, which is 12.
• Such 8 numbers are possible which are completely divisible by 12. They are 12, 24, 36, 48, 60, 72, 84 and 96.
Q 6. LCM AND HCF
Five persons fire bullets at a target at an interval of 6, 7, 8, 9 and 12 seconds respectively. The number of times they would fire the bullets together at the target in an hour is
The five persons will together fire at the target at an interval of 'r second where
t = LCM (6, 7, 8, 9. 12)
LCM (6, 7, 8, 9, 12) = 23 x 32 x 7 = 504
⸫ t = 504 s
Number of seconds in an hour = 3600
⸫ Number of times the five persons would fire the bullets
together at the target in an hour =
(where
Q 46. LCM AND HCF
A bell rings every 18 minutes. A second bell rings every 24 minutes. A third bell rings every 32 minutes. If all the three bells ring at the same time at 8 o'clock in the morning, at what other time will they all ring together?
Interval of first bell ring = 18 min
Interval of second bell ring = 24 min
Interval of third bell ring = 32 min
Now interval after which they will ring together = LCM (18 min, 24 min and 32 min)
= 4 x 3 x 3 x 2 x 2 x 2 = 288 min
⸫ Bells will ring after 288 min =
Now, if they ring at 8 O’ clock in the morning, then they will ring again at = 8 + 4 h 48 min = 12 : 48 h
Hence. option (b) is correct.
Q 40. LCM AND HCF
Three persons start walking together and their steps measure 40 cm, 42 cm and 45 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
It’s a simple LCM problem. Take the LCM of 40, 42 and 45 to know at what point will they find a common multiple. It is 2520 cm i.e. 25 m 20 cm.