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22 Jul 2023, 08:19
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Hi all,
I was going through the number properties chapter and I stumbled upon something that's a bit confusing to me. For instance, in 3^123/4 the remainder is easy to find out because, the remainder pattern is repeating: 3-1-3-1 etc. (3^1/4 : remainder 3, 3^2/4 : remainder is 1, so on and so forth) So it’s a straight forward way to find out 3^123, since the exponent is an odd number, then the remainder must be 3. However, in a case where the divisor is 7, the remainder repeating pattern is : 2-4-1-2-4-1 etc. How to find out the remainder for a base raised to a larger exponent, like 2^15/7 or 2^131/7 etc?
When 2^1/7 leaves a remainder of 2, 2^3/7 leaves a remainder of 1, 2^5/7 leaves a remainder of 4, my question is, when bases raised to odd exponents give different remainder values, how to figure out the pattern? Or even calculate remainders for questions like 2^13/7 or 2^15/7 or 2^131/7 etc?
I was going through the number properties chapter and I stumbled upon something that's a bit confusing to me. For instance, in 3^123/4 the remainder is easy to find out because, the remainder pattern is repeating: 3-1-3-1 etc. (3^1/4 : remainder 3, 3^2/4 : remainder is 1, so on and so forth) So it’s a straight forward way to find out 3^123, since the exponent is an odd number, then the remainder must be 3. However, in a case where the divisor is 7, the remainder repeating pattern is : 2-4-1-2-4-1 etc. How to find out the remainder for a base raised to a larger exponent, like 2^15/7 or 2^131/7 etc?
When 2^1/7 leaves a remainder of 2, 2^3/7 leaves a remainder of 1, 2^5/7 leaves a remainder of 4, my question is, when bases raised to odd exponents give different remainder values, how to figure out the pattern? Or even calculate remainders for questions like 2^13/7 or 2^15/7 or 2^131/7 etc?
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26 Jul 2023, 05:04
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Yes, I do have a similar query while doing the Number Properties chapter. JeffTargetTestPrep could you pls reply to the above-mentioned ques?
Thanks in Advance!
Thanks in Advance!
26 Jul 2023, 13:42
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kelsier1569
For example, let’s determine the remainder of the division 2^131/7.
First, we have to explore the repeating remainder pattern:
2^1/7 = 0 remainder 2
2^2/7 = 0 remainder 4
2^3/7 = 1 remainder 1
2^4/7 = 2 remainder 2
We can stop at the first repeating number and conclude that the repeating remainder pattern of 2^k/7 is 2-4-1.
In other words, every third power of 2 will have a remainder of 1 after division by 7.
Since we are looking for the remainder of the division 2^131/7, we find the closest number to 131 that is a multiple of 3. We see that number is 132.
If the question were about the remainder of the division 2^132/7, then our answer would be 1.
However, 131 is one less than 132, so we have to move back one place from 1 in the remainder pattern 2-4-1.
Therefore, the remainder of the division 2^131/7 is 4.
