please explain this mcq- According to Euclid's division of Lemma, for a and 5 if a=5q+r is unique then r = ...... is not possible.
A. 0
B. 2
C. 4
D. 6
To solve this multiple-choice question, we need to understand Euclid's division lemma and apply it to the given information.
Euclid's division lemma states that for any two positive integers, a dividend (a) and a divisor (d), there exist unique integers, quotient (q) and remainder (r), such that a = dq + r, where 0 ≤ r < d.
In this case, we are given that a = 5q + r, where 'a' represents a positive integer and '5' is the divisor. To determine the value of 'r' that is not possible, we need to examine the range of values for 'r' based on Euclid's division lemma.
Since the divisor is 5, we know the range of values for 'r' lies between 0 and 4 (exclusive).
Now, let's evaluate the given options:
A. 0: This is a possible value for 'r'. For example, if a = 5 and q = 1, then a = 5q + r = 5(1) + 0 = 5.
B. 2: This is also a possible value for 'r'. For example, if a = 7 and q = 1, then a = 5q + r = 5(1) + 2 = 7.
C. 4: This is again a possible value for 'r'. For example, if a = 14 and q = 2, then a = 5q + r = 5(2) + 4 = 14.
D. 6: This is outside the range of possible values for 'r'. According to Euclid's division lemma, if a = 5q + r, then the remainder 'r' should be between 0 and 4. Therefore, 'r = 6' is not possible.
Hence, the correct answer is D. 6.
The Lemma says that for any pair, a,b we can find q,r such that
a = bq+r
The above equation is just
a ≡ r (mod 5)
There is no unique value of a which will determine r.