Problem Solving

BTGModeratorVI wrote: ↑

Sat Aug 08, 2020 7:09 am

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

Answer: C
Source: Magoosh

Given: x + 2 is divisible by 10
In other words x + 2 is a multiple of 10, which means we can write: x + 2 = 10k for some integer k

We want to know what the remainder will be when we divide x² + 4x + 9 by 10.

x² + 4x + 9 = x² + 4x + 4 + 5
= (x +2)(x +2) + 5
= (10k)(10k) + 5
= 100k² + 5
So now we want to know what the remainder will be when we divide 100k² + 5 by 10.

We can already see that 100k² [aka (10)(10k²)] is a MULTIPLE of 10
So, 100k² + 5 is 5 greater than a multiple of 10
So, the remainder will be 5 when we divide 100k² + 5 by 10

Answer: C

Cheers,
Brent

BTGModeratorVI wrote: ↑

Sat Aug 08, 2020 7:09 am

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

Answer: C

Solution:

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