1) Determine the lowest natural number which when divided by 16,28,40 and
77 leaves remainder 8 in each case.
Solution : We know that the lowest number divisible by 16,28,40 and 77 is
their LCM. Therefore the required number must be 8 more than their LCM.
LCM of 16,28,40 and 77 (use any of the above method)
Hence the required number = ( 6160 + 8) = 6168.
2) Determine the two numbers nearest to 10,000 which are exactly divisible by 2,3,4,5,6 and 7.
Solution : The smallest number which is exactly divisible by 2,3,4,5,6 and 7 is their LCM.
LCM of 2,3,4,5,6 and 7 = 420
The number nearest to 10,000 and exactly divisible by each given numbers
should also be exactly divisible by their LCM 420.
So divide 10,000 by 420.
We find the remainder is 340.
Number just less than 10,000 and exactly divisible by 420
= 10000 -340 =9660
Number just greater than 10,000 and exactly divisible by 420
= 10000 + (420 - 340) = 10080
Hence, the required numbers are 9660 and 10080.
__________________________________________________________________ Some Properties of GCF and LCM
1) The GCF of given numbers is not greater than any of the numbers.
2) The LCM of given numbers is not less than any of the given numbers.
3) The GCF of two co-prime numbers is 1.
4) The LCM of two or more co-prime numbers is equal to their product.
5) The GCF of given numbers is always a factor of their LCM.
6) The product of LCM and GCF of two number is equal to the product of the given numbers. That is, if a and b are the two numbers, then
a x b = GCF x LCM Example : Given that GCF of two numbers is 16 and their product is 6400, determine their LCM.