If (z + 3)/5 is an integer, what is remainder when z is divided by 5?

Assume the answer choices as the value of z

A) 2; (2+3)/5 = 1 satisfies the equation
B) 3; (3+3)/5 = 1.2 Not an integer
C) 4; (4+3)/5 = 1.4 Not an integer
D) 5; (5+3)/5 = 1.6 Not an integer
E) 6; (6+3)/5 = 1.8 Not an integer

Ans A
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Answer = A. 2

Let z = 2, gives \(\frac{2+3}{5} =\)integer

For \(\frac{2}{5}\), remainder = 2

(z+3)/5 is an integer, which means that remainder must be a multiple of 5.
Eq can be written as z/5 + 3/5,
3/5 gives a remainder of 3, which means that z must be 5n+2, where n is an integer.
Answer is A.

From the question we can see that if (z+3)/5 is an integer it means that z+3 should be a multiple of 5. Hence z=2 or z=7 or z=12 and so on. Hence if,for example, z=7 then (7/5) leaves a remainder of 2. If z=12 then (12/5) leaves a remainder of 2 again. Hence answer A.

(z+3)/5 = I
z+3=5I
adding two on both sides we have
z+3+2 = 5I+2
z+5=5I+2
z=5I-5+2
=5(I-1)+2

thus z when divided by 5 will always gives a remainder of 2.

(z + 3)/5 is an Integer.
Lets say K+1
(z + 3)/5 = k+1
z+3 = 5k+5
z=5k+5-3
z=5k+2

So when z is divided by 5 reminder is 2.

Z is an integer that needs 3 to be divisible by 5
in that case, whatever number Z is, the remainder will always be 2.

Since (z + 3)/5 is an integer , \(\frac{(z + 3)}{5}\) is an integer must be divisible by 5

The least possible value of z must be 2

\(\frac{2}{5}\) - reminder is 2 ( Answer choice A)

Hence the correct answer is A
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Solution:

The question states that z+3/5 is an integer, therefore we know that z+3 is a multiple of 5. Now to solve this question we can plug in a value for z+3 to arrive at the solution.
Lets assume z+3 to be any multiple of 5. Let say z+3 = 15, then z = 12. The question asks what is the remainder when z is divided by 5. When 12/5 we know the remainder is 2. If we plug in any other multiples of 5 for z+3, then too we will get a remainder of 2 when z is divided by 5. Therefore our answer is A.
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(z + 3)/5 = Integer......Then z could be 2,7,12,17...etc

In the above.........z /5 .......reminder is 2

Answer: A

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