LCM and HCF 1
The least number of five digits which is exactly divisible by 12, 15 and 18, is:
Least number of 5 digits is 10,000 L.C.M. of 12, 15 and 18 is 180.
On dividing 10000 by 180, the remainder is 100.
∴ Required number
= 10000 + (180 – 100) = 10080.
The greatest number of four digits which is divisible by 15, 25, 40 and 75 is:
Greatest number of 4 digits is 9999. L.C.M. of 15, 25, 40 and 75 is 600.
On dividing 9999 by 600, the remainder is 399.
= (9999 – 399) = 9600.
The least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3 is:
L.C.M. of 5, 6, 4 and 3 = 60.
On dividing 2497 by 60, the remainder is 37.
∴ Number to be added
= (60 – 37) = 23.
Find the maximum number of students among whom 429 mangoes and 715 oranges can be equally distributed.
Required number
= HCF of 429 and 715 = 143
Find the greatest number that will divide 115, 149 and 183 leaving remainders 3, 5, 7 respectively.
= HCF of (115 – 3), (149 –5) and (183 – 7)
= HCF of 112, 144 and 176 = 16
Find the greatest number which when subtracted from 3000 is exactly divisible by 7, 11, 13.
= 3000 – LCM of 7, 11, 13
= 3000 – 1001 = 1999
The L.C.M. of two number is 630 and their H.C.F. is 9. If the sum of numbers is 153, their difference is
Let numbers be x and y.
Product of two numbers = their (LCM × HCF)
⇒ xy = 630 × 9
Also, x + y = 153 (given)
since x – y
⇒
Product of two co-prime numbers is 117. Their L.C.M. should be:
H.C.F of co-prime numbers is 1.
So, L.C.M. = 117/1 = 117.
What is the smallest number which when increased by 5 is completely divisible by 8, 11 and 24?
= LCM of ( 8, 11, 24 ) – 5
= 264 – 5 = 259
If the L.C.M and H.C.F. of two numbers are 2400 and 16, one number is 480; find the second number.
Product of numbers
= (LCM × HCF)
⇒ 480 × second number = 2400 × 16
⇒ second number = 80
The LCM of two numbers is 4800 and their HCF is 160. If one of the numbers is 480, then the other number is:
Product of numbers = HCF × LCM
⇒ The other number
=
The LCM of two numbers is 280 and their ratio is 7: 8. The two numbers are:
Let the numbers be 7x and 8x.
⇒ Their HCF = x
Now, LCM × HCF = Product of Numbers
i.e. 280 × x = 56x²
⇒ x = 5
Hence, the numbers are 35 and 40.
The LCM and HCF of two numbers are 84 and 21, respectively. If the ratio of two numbers be 1: 4, then the larger of the two numbers is:
Let the numbers be x and 4x.
Then, 84 × 21 = x × 4x
⇒ 4x² = 1764
⇒ x² = 441
⇒ x = 21
⇒ 4x = 4 × 21 = 84
Thus the larger number = 84
How many numbers, between 1 and 300 are divisible by 3 and 5 together?
LCM of 3 and 5 = 15
∴ 300/15 = 20 numbers
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