show that the square of any two positive integer is of the form 8-k + 1 for some Integer k
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The square of an odd
positive integer is of the
form 8q+1
1) Show that the square of an odd positive
integer is of the form 8q+1, for some integer
q.
2) Prove that the square of any positive
integer of the form 5q+1 is of the same
form.
3) A mason has to fit a bathroom with square
marble tiles of the largest possible size. The
size of the bathroom is 10ft by 8ft. What
would be the size in inches of the tile
required that has to be cut and how many
such tiles are required?
4) Determine the number nearest to 110000
but greater than 100000 which is exactly
divisible by 8,15 and 21.
PLEASE GIVE EXPLANATION FOR ALL
QUESTIONS.
Solution:
By Euclids division lemma, given
positive integers a and b, there exist
unique integers q and r satisfying a = bq
+ r, 0 ≤ r < b.
positive integer is of the
form 8q+1
1) Show that the square of an odd positive
integer is of the form 8q+1, for some integer
q.
2) Prove that the square of any positive
integer of the form 5q+1 is of the same
form.
3) A mason has to fit a bathroom with square
marble tiles of the largest possible size. The
size of the bathroom is 10ft by 8ft. What
would be the size in inches of the tile
required that has to be cut and how many
such tiles are required?
4) Determine the number nearest to 110000
but greater than 100000 which is exactly
divisible by 8,15 and 21.
PLEASE GIVE EXPLANATION FOR ALL
QUESTIONS.
Solution:
By Euclids division lemma, given
positive integers a and b, there exist
unique integers q and r satisfying a = bq
+ r, 0 ≤ r < b.
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