multiplicative climax

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Your solution: -2*86 = 172

- 3 (remainder, no factor)

- 4*43 = 172

- 5 (remainder, no factor)

- 6 (remainder, no factor)

- 7(remainder, no factor)

- 8(remainder, no factor)

- 9 (remainder, no factor)

- 10 (remainder, no factor)

- 11 (remainder, no factor)

- 12(remainder, no factor)

13 (NF)

- 14(NF)

- 15 (NF)

- 16(NF)

- 43…

Therefore, all factors of 172 are (2 and 86), (4 and 43) and of course the 1:1 ratio, (1 and 172)

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Given Solution:

`a** If the number isn't too big we can simply start dividing by primes, beginning with the smallest:

If we divide 172 by 2 we get 86, so 2 and 86 are factors.

If we divide 172 by 3 we get 57 with a remainder so 3 isn't a factor.

If we divide 172 by 4 we get 43, so 4 and 43 are factors.

If we divide 172 by 5 we get 34 with a remainder so 5 isn't a factor.

If we divide 172 by 6 we get 28 with remainder so 6 isn't a factor.

If we divide 172 by 7 we get 24 with a remainder so 7 isn't a factor.

If we divide 172 by 8 we get 21 with remainder so 8 isn't a factor.

If we divide 172 by 9 we get 19 with a remainder so 9 isn't a factor.

If we divide 172 by 10 we get 17 with a remainder so 10 isn't a factor.

If we divide 172 by 11 we get 15 with a remainder so 11 isn't a factor.

If we divide 172 by 12 we get 14 with a remainder so 12 isn't a factor.

If we divide 172 by 13 we get 13 with a remainder so 13 isn't a factor.

If we were to divide 172 by any number greater than 13 the result would be less than 13. We've already divided by every whole number less than 13 so we aren't going to find anything new by dividing by numbers greater than 13.

Our factors are 2, 86, 4 and 43, as well as 1 and the number 172 itself.

A method which is often quicker if the prime factorization contains a large number of factors is to list every prime factor, every product of two prime factors, every product of three prime factors, etc.:

From the Prime Factorization 172 = 2 * 2 * 43 you find that the factors include:

Each prime factor: 2 and 43

Each product of two prime factors: 2 * 2 = 4 and 2 * 43 = 86

The number itself and 1: 1 and 172.

This method is quicker and more reliable than dividing by every possible number (what would you do with 5,668,725, for example?). **

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Self-critique (if necessary): Can one generally assume that, when determining factors, when one reaches a divisor whose quotient, as the whole number, is equal to the divisor, we’ve reached the “multiplicative climax?”

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Self-critique Rating:

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I really like the way you put that.
That is so.
You don't have to understand, or even read what follows but your description deserves further comment:
A quick proof that it's so:
If the whole-number part of n / p is equal to p, then n / p is less than p + 1 (e.g., 29 / 5 = 5.8, which is less than 5 + 1, which would be 6).
So the condition you state can be written as
n / p < p + 1.
A little algebra (specifically, multiplying both sides by p and then dividing both sides by p + 1) tells us that this is equivalent to
n / (p + 1) < p.
This tells us that if we divide by the next-higher number the result will be less than the divisor, which is the condition that characterizes the 'multiplicative climax', as you so nicely describe it.
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