Euclid Division Lemma
Euclid Division Lemma
Euclid Division Lemma:
If a is an integer and b is a positive integer then there exist two integers q and r such that
q and r are called quotient and remainder respectively when a is divided by b.
For instance, if a = -86, b = 5 then -86 =5(-18) + 4
Thus, q = -18 and r = 4
Note that q and r in the above statement are unique.
Example: Show that any positive even integer is of the form 8q, 8q +2, 8q + 4, 8q + 6, where q is some integer.
Solutions:
Let a and b = 8 be two positive integers where a is even.
Applying division lemma a = 8q + r where 0 r <8
So, r can take any of the values 0, 1, 2, 3, 4, 5, 6, 7
Therefore, a = 8q, 8q + 1, 8q + 2, 8q + 3, 8q + 4, 8q +5, 8q + 6,8q + 7, 8q + 8
Since, a is even.
Therefore, a can take values 8q, 8q + 2, 8q + 4, 8q + 8 since they can expressed as multiples of 2.
So, a will not take value 8q + 1, 8q + 3, 8q + 5, 8q + 7. as they all represent odd numbers.
On applying division lemma to 17 and, 7, You get 17 = a X 7 + 3. What is the value of a? | |||
| Right Option : B | |||
| View Explanation | |||
On applying division lemma to 36 and 5, you get | |||
| Right Option : A | |||
| View Explanation | |||
On applying division Lemma to 25 and X, You get 25 = 8X + 1 . What is the value of x? | |||
| Right Option : C | |||
| View Explanation | |||
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