If x is a positive integer and x + 2 is divisible by 10, what is the remainder when x^2 + 4x + 9 is divided by 10?

A. 1
B. 3
C. 5
D. 7
E. 9
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\(x^2 + 4x + 9\) can be written as
\(x^2 + 4x + 4+5=(x+2)^2+5\)
Since x+2 is divisible by 10, hence
Rem(\(\frac{(x+2)^2+5}{10}\))=Rem(\(\frac{(x+2)^2}{10}\))+Rem(\(\frac{5}{10}\))=0+5=5

Ans. (C)
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Least value of x to be divisible by 10 is x = 8

Now, \(\frac{x^2 + 4x + 9}{10} = \frac{8^2 + 4*8 +9}{10}=\frac{105}{10}\)

Thus, Remainder will be 5, Answer must be (C)
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Since x + 2 is divisible by 10, then x + 2 could equal 10, and hence, x could equal 8. (Note, that x + 2 could also equal 20, or 30, or any positive multiple of 10).

We’ll choose x = 8, and we have:

8^2 + 4(8) + 9 = 64 + 32 + 9 = 105

Since 105/10 = 10 R 5, the remainder is 5.

Alternate Solution:

Let’s write x^2 + 4x + 9 = x^2 + 4x + 4 + 5 = (x + 2)^2 + 5.

Since x + 2 is divisible by 10, (x + 2)^2 is also divisible by 10. Hence, the remainder when x^2 + 4x + 9 = (x + 2)^2 + 5 is divided by 10 is 5.

Answer: C
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