Question about how 8*8*8*8 = 4 Find all integers for which this statement is true, modulo n.
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Question about how 8*8*8*8 = 4 Find all integers for which this statement is true, modulo n.

[From: ] [author: ] [Date: 11-05-16] [Hit: ]
Stevie Wonder mentions the equation 8*8*8*8=4. Find all integers n for which this statement is true, modulo n.The answer in the back of the book reads the statement is true for any divisor of 8^4 - 4=4091.Im just confused........
In the cut "as" from the songs in the key of Life, Stevie Wonder mentions the equation 8*8*8*8=4. Find all integers n for which this statement is true, modulo n.

The answer in the back of the book reads "the statement is true for any divisor of 8^4 - 4=4091. I'm just confused....it doesn't make sense to me. The question says "find all integers"

Any help would be greatly appreciated!

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In modulo arithmetic, congruence (not equality) is based on the remainder, when you divide a number by the base of the modulus.

For example, in modulo 10, you would only keep the last digit, as that is the remainder, when you divide by 10

7 * 5 is congruent to 5 (modulo 10)
because 7 * 5 = 35 and 35/10 leaves a remainder of 5

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Normally we should begin with numbers higher than 4 as a base, as one does not have remainders higher that the base.

8 is congruent to 3 (modulo 5)
8*8 is therefore congruent to 3*3 = 9, congruent to -1
(8*8)*(8*8) is congruent to (-1)^2 = 1 (mod 5)

8 is congruent to 2 modulo 6
8*8 is congruent to 4 (mod 6), same as -2 (mod 6)
8*8*8*8 is congruent to (-2)^2 = 4 (mod 6)
So we know that there is one answer at n=6

But, of course, some will complain that we cannot really take n to be less than 8, so we have to keep on looking.

Using the method:

8^4 - 4 = 4092 (not 4091)
The reason that this method works is that 4092 is 4 less than 8*8*8*8. therefore, if we divide
8*8*8*8 by 4092, the remainder will be 4.
If we divide 8*8*8*8 by any divisor of 4092, the remainder will be congruent to 4 (if n is 4 or less, then of course, 4 itself will be congruent to a smaller number).

The prime factors of 4092 are:
2*2*3*11*31

Let's try a few, just for the fun of it:
n = 31
8*8 = 64 = 2*31 + 2 = 2 (mod 31)
8*8*8*8 = (2)^2 = 4 (mod 31)

n = 2*11 = 22
8*8 = 64 = 66 - 2 = -2 (mod 22)
8*8*8*8 = (-2)^2 = 4 (mod 22)
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