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353 has primitive roots because it is a Prime number. (Not all numbers have primitive roots) It has 160 primitive roots. Generally however when we talk about THE primitive root we are talking about the lowest. So, starting at 2,3,4,5, ...... create a spreadsheet where the columns are the numbers being tested and the rows contain the powers of these numbers modulo 353 up to 176 (=352/2).

The first column not containing a 1. is the lowest primitive root.

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Q: How do you find the primitive root of 353?
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What is the square root of 353?

18.788294the square root of 353 times the square of 353 equals 353


How do you find the primitive roots of 31?

You need to know the base for the modulo arithmetic. If the base is the integer p, then 31 can be expressed as n*p + r, or r is the remainder when 31 is divided by p. r is also called the primitive root of 31 modulo p. The primitive root of 31, by itself, has no meaning. It is only in relation to p that it has a meaning and a value. p = 2: primitive root = 1 p = 3: primitive root = 1 p = 4: primitive root = 3 p = 5: primitive root = 1 p = 7: primitive root = 3 p = 8: primitive root = 7 etc.


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The root word "archaeo" means ancient or old. It is commonly used in words related to ancient artifacts, cultures, or practices.


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first (Latin primus)examples: primary, primitive, prime


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Expressed as a surd, the square root of 1412 is equal to 2 sqrt(353). Expressed as a number rounded to two decimal places, it is approximately equal to 37.58.


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