LCM and HCF
LCM
Lowest common multiple of two or more given numbers is the least or the smallest number which is exactly divisible by each of them
Quick method to find the LCM
- Put down the numbers in a row by separating them with commas
- We first divide the numbers by 1st prime number i.e. 2 and continue with next prime number until we have at least 2 numbers which are exactly divisible by the prime numbers.
- We set down the quotients and indivisible numbers in a row below the first and so on. Repeated process of division gives us a row of numbers which are prime to one another. The product of us all the divisors and the remainders in the last row gives us the required LCM.
HCF
Highest common multiple or greatest common divisor(GCD) of two or more numbers is the greatest number which divides each of the numbers exactly
Quick method to find the HCF of 2 numbers
- We divide the larger number by the smaller number and find out the remainder.
- Then divide the first divisor by the remainder and find the second remainder.
- Then divide the second divisor by the second remainder.
- We repeat the process till no remainder is left. The divisor is our required HCF
Important tips:
- Product of 2 numbers: HCF x LCM
- Product of n numbers: (LCM of n numbers) x (Product of the HCF of each possible pair)
- If HCF(a,b) is H1 and HCF(c,d) is H2, then HCF(a,b,c,d) is HCF(H1, H2)
- LCM is always a multiple of HCF i.e. LCM = (a number) x HCF
- To find the greatest number that will exactly divide x, y, z; just find the HCF of these numbers
- To find the greatest number that will divide x, y, z leaving remainders a, b and c respectively; then find the HCF of (x-a), (y-b) and (z-c)
- Find the least number which is exactly divisible by x, y and z; then find their LCM
- Find the least number which when divided by x, y and z leaving remainders a, b and c respectively, then the required number= (LCM of x, y and z) – K ; where K= (x-a) = (y-b) = (z-c)
Now try to solve these 10 questions as fast as possible (5-7 min)
1. Find the greatest number which divides 10997 and 14139 exactly?
(a) 1571 (b) 7650
(c) 571 (d) 671
2. Find the HCF of 35/12, 49/30 and 21/20.
(a) 7/36 (b) 5/49
(c) 7/60 (d) 13/55
3. Find the least number which is exactly divisible by 2, 3, 4, 5, 6 and 7.
(a) 830 (b) 420
(c) 210 (d) 460
4. Find the largest number of 4 digits which is completely divisible by 48, 40 and 64.
(a) 9960 (b) 9880
(c) 9600 (d) 9900
5. Three numbers are in the ratio 1:2:3 and their HCF is 12. The numbers are:
(a) 12, 24, 36 (b) 11, 22, 33
(c) 12, 18, 24 (d) 5, 10, 15
6. Find the least number divisible by 6, 8, 9 and 12 which is a perfect square.
(a) 196 (b) 144
(c) 256 (d) 324
7. What is the least numbers of soldiers that can be drawn up in troops of 12, 15, 18 and 20 and also in the form of a solid square?
(a) 900 (b) 400
(c) 1600 (d) 2500
8. Find the side of the largest possible square slabs which can be paved on the floor of a room 2m 50cm long and 1m 50cm broad. Also find the number of such slabs to pave the floor.
(a) 50, 20 (b) 30, 15
(c) 25, 20 (d) 30, 20
9. Find the greatest number by which if we divide 120, 153 and 252, then in each case the remainder is the same.
(a) 7 (b) 11
(c) 10 (d) 5
10. A heap of pebbles when made up into groups of 32, 40, 72 leave the remainders 10, 18 and 50 resp. Find the least number of pebbles on the heap?
(a) 1344 (b) 1564
(c) 1418 (d) 1600
SOLUTIONS
Q | Ans | Q | Ans |
1 | a | 6 | b |
2 | c | 7 | a |
3 | b | 8 | a |
4 | c | 9 | b |
5 | a | 10 | c |
Time taken
Within 5 min: EXCELLENT
5-7 min: YOU CAN DO BETTER
More than 7 min: YOU NEED TO WORK HARD
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