Euclid'S division Lemma
Euclid 's division lemma says that for any given positive integers a and b, there exist unique whole numbers q and r satisfying
a=bq+r, 0<r<b
Try your self
1. Show that any positive odd integer is of the form 6q+1,or 6q+3 or 6q+5 where q is some integer .[Hint b=6, r 0,1,2,3,4,5. (0<r<b)]
Answer Let a be any positive integer and b= 6. Then , by Euclid'S Division Lemma a = bq+r ,0<r<b, So possible values of r are 0,1,2,3,4 or 5
If r=0 , then a = 6q + 0 = 6q (+ve even integer )
If r=1 , then a = 6q + 1 (+ve odd integer )
If r=2 , then a = 6q + 2 (+ve even integer )
If r=4 , then a = 6q + 4 (+ve even integer )
If r=5 , then a = 6q + 5 (+ve odd integer )
Hence, these expressions of numbers are odd numbers. And therefore, any odd integer can be expressed in the form 6q+ 1, or 6q + 3, or 6q + 5.
2. If sections of these two classes are to be made, where number of students should be same in each section then find the maximum number of the sections. [ Hint: HCF ( 576, 448 ) =? ]
Answer Using Euclid'S Division algorithm :
Step 1 : 576 = 448 × 1 + 128
Step 2 : 448 = 128 × 3 + 64
Step 3 : 128 = 64 × 2 + 0
As remainder is 0 , So we can't proceed any further Hence the division 64 is the HCF of 576 and 448.
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