Here is the correct question, along with the answer choices:
If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?
A. 0
B. 1
C. 2
D. 3
E. 4
We can plug in any positive integer for x.
If x=1, then 7^(12x+3) + 3 = 7^15 + 3.
Question rephrased: What is the remainder when 7^15 + 3 is divided by 5?
When an integer is divided by 5, the remainder is determined by the UNITS DIGIT.
Thus, we need to know the units digit of 7^15 + 3.
When an integer is raised to consecutive powers, the resulting units digits repeat in a CYCLE:
7¹ --> units digit of 7.
7² --> units digit of 9. (Since the product of the preceding units digit and 7 = 7*7 = 49.)
7³ --> units digit of 3. (Since the product of the preceding units digit and 7 = 9*7 = 63.)
7� --> units digit of 1. (Since the product of the preceding units digit and 7 = 3*7 = 21.)
From here, the units digits will repeat in the same pattern: 7, 9, 3, 1.
The units digit repeat in a CYCLE OF 4.
Implication:
When 7 is raised to a power that is a multiple of 4, the units digit will be 1.
Thus, 7^12 has a units digit of 1.
From here, the cycle of units digits will repeat:
7^13 --> units digit of 7.
7^14 --> units digit of 9.
7^15 --> units digit of 3.
Since 7^15 has a units digit of 3, 7^15 + 3 has a units digit of 6.
When an integer with a units digit of 6 is divided by 5, the remainder in each case is 1:
16/5 = 3 R1.
26/5 = 5 R1.
36/5 = 7 R1.
Thus, when 7^15 + 3 is divided by 5, the remainder will be 1.
The correct answer is
B.
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