(1) Product of two numbers = Their H.C.F. x Their L.C.M.= axb= H.C.F.(a,b) x L.C.M(a,b)
(2) H.C.F. of given numbers always divides their L.C.M.
(3) H.C.F. of given fractions = (H.C.F. of Numerator)/(L.C.M. of Denominator)
(4) L.C.M. of given fractions = (L.C.M. of Numerator)/(H.C.F. of Denominator)
(5) If d is the H.C.F. of two positive integer a and b, then there exist unique integer m and n, such that d = am + bn
(6) If a = bxm +n then H.C.F.( a, b) = H.C.F.(b , n) , where m,n are integers
(7) The HCF of any set of numbers is smaller than or equal to the smallest number
(8) The LCM of any set of numbers is greater than or equal to the largest number.
(9) Two numbers are co-prime if their H.C.F. is 1
(10) Find the “Greatest number” that will exactly divide a,b and c,
So required number will be = H.C.F. of ( a,b,c)
(11) Find the “Greatest number” that will divide p,q and r , and leaving remainder x,y,and z
So required number will be H.C.F. of { (p-x),(q-y) and (r-z)}
(12) Find the “Least number” that will exactly divisible by a,b and c
So required number will be L.C.M. of (a,b,c)
(13) Find the “Least number” when divide by p,q and r leaves the remainder x,y and z respectively and it must be observed that (p-x) = (q-y) = (r-z) = N
So required number will be L.C.M. of ( p,q,r) – N
(14) Find the “Least number” when divide by p, q and r leaves the same remainder “n”
So required Number will be L.C.M. of ( p,q,r) + n
(15) Find the “Greatest Number “ that will divide p , q and r leaving the same remainder in each case
So required number will be H.C.F. of (p-q) , (q-r) and (r-p)
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