(11) 196 and 38220
(iii) 867 and 255
2. Show that any positive odd integer is of the form 6q+1, or 6q+3, or 69 +5, where is
some integer.
3. An army contingent of 616 members is to march behind an army band of 32 members in
a parade. The two groups are to march in the same number of columns. What is the
maximum number of columns in which they can march?
4. Use Euclid's division lemma to show that the square of any positive integer is either of
the form 3m or 3m + 1 for some integer m.
[Hint : Let x be any positive integer then it is of the form 39,34 + 1 or 34+2. Now square
each of these and show that they can be rewritten in the form 3m or 3m + 1.]
5. Use Euclid's division lemma to show that the cube of any positive integer is of the form
9m, 9m + 1 or 9m + 8.
Answers
- (1).196 and 38220
- 38220 > 196
- than by Euclid division lemma
- 38220 = 196 × 196 + 0
- H.C.F (196 ,38220)=196
(2).867and255
867 > 255
than by Euclid division lemma
867 =255×3+102
255 = 102×2+51
102 = 51×2+0
H.C.F(867,255) =51
2.) solution. let a be any positive odd integer and b=6
than by Euclid division algorithm
a=bq +r,0< r > b
a= 6q + r,0< r > 6
Here,total possible values of r = 0,1,2,3,4,5 if r= other a=rq
r=1 ......... a =6q + 1
r =2......... a= 6q+2
r =3...........a= 6q +3
r = 4.........a= 6q+4
r = 5........a =6q +5
6q +2,6q+4
if a=6q+2,6q+4 that is can be divisible by 2
since is odd integer and cannot be of the form 6q+2,6q+4
therefore any positive odd integer is of the form 6q +1, 6q+3,6q+5
3). solution..The maximum number of column in which take on much is the H.C.F.of 616 ,32
616>32
than by Euclid division algorithm
616= 32 × 19 +8
32 = 8 × 4 + 0
H.C.F.( 616,32 )=8
sorry last 2 nhi kiye he kiuki power problem ho rhi he