Problem Solving

I can't seem to grasp this answer..even with the answer...can someone help? thanks!

If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

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ANSWER:


To find the remainder when a number is divided by 5, all we need to know is the units digit, since every number that ends in a zero or a five is divisible by 5.

For example, 23457 has a remainder of 2 when divided by 5 since 23455 would be a multiple of 5, and 23457 = 23455 + 2.

Since we know that x is an integer, we can determine the units digit of the number 7^(12x+3) + 3. The first thing to realize is that this expression is based on a power of 7. The units digit of any integer exponent of seven can be predicted since the units digit of base 7 values follows a patterned sequence:

Units Digit = 7


Units Digit = 9


Units Digit = 3


Units Digit = 1
Since we see that the pattern repeats itself every 4 integer exponents.

The question is asking us about the 12x+3 power of 7. We can use our understanding of multiples of four (since the pattern repeats every four) to analyze the 12x+3 power.

12x is a multiple of 4 since x is an integer, so 7 12x would end in a 1, just like 74 or 78.
7 12x+3 would then correspond to 73 or 77 (multiple of 4 plus 3), and would therefore end in a 3.

However, the question asks about 712x+3 + 3.
If 7 12x+3 ends in a three, 7 12x+3 + 3 would end in a 3 + 3 = 6.

If a number ends in a 6, there is a remainder of 1 when that number is divided by 5.

The correct answer is B.