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Number Properties

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Source: — Problem Solving |
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by aplavakarthik » Mon Sep 26, 2011 8:54 am
7 when multiplied by 7 follows a sequence to the end which repeats.

7=7
7^2=49
7^3=343
7^4=2041
7^5=7

so 7,9,3,1 keeps on repeating. given x is a positive integer. let it be 1 7^15 will end with 3 and +3 will give u a 6 to last. so 1 will be the remainder.
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by tpr-becky » Mon Sep 26, 2011 2:55 pm
In a problem such as this that seems to require lots of time consuming calculation it is best to look for a pattern in the numbers:

7^1 = 7
7^2 = 49
7^3 = 343
7^4 = 2401
7^5 = 16807

Then you can see the pattern in the units digit. we know that when you divide by 5 the remainder will be any units digit between 1 and 4 or 5 minus the units digit between 6 and 9.

So we need to find the units digit. There is no definition for x so we can assume it can be 1 - then you are looking for 7^15 + 3 - use the pattern above to simply count to 15 and discover that the units digit of 7^15 will be 3 therefore the units digit of the equation will be 6. when you divide a number with a 6 as a units digit by 5 you will get a remainder of 1. Thus the answer is 1.
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by sparkle6 » Mon Sep 26, 2011 9:27 pm
But if you take x = 2 or 3, etc... doesn't the answer change?
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by Anurag@Gurome » Mon Sep 26, 2011 9:32 pm
sparkle6 wrote: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

[spoiler]Answer: B.[/spoiler]

The units digits of the powers of 7 has a cycle of 7, 9, 3, 1, 7, 9, ...

Hence, units digit of 7^(Some multiple of 4 + 3) = units digit of 7^3 = units digit of 343 = 3

Hence, units digit of 7^(12x + 3) + 3 = 3 + 3 = 6

Hence, required remainder = 6 - 5 = 1

The correct answer is B.
Anurag Mairal, Ph.D., MBA
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