Posts tagged 'prime-to-one-another'

G.249 numbers prime to one another

Numbers prime to one another are those which are:term:measured <G.253> by an:term:unit <G.247> alone as a common measure.

G.250 unequal numbers set

Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers is prime to one another.

G.251 numbers prime find

Given two numbers not prime to one another, to find their greatest common measure.

G.252 numbers prime find

Given three numbers not prime to one another, to find their greatest common measure.

G.255 number part parts

Any number is either a part or parts of any number, the less of the greater.

G.275 numbers prime least

Numbers prime to one another are the least of those which have the same ratio with them.

G.276 least numbers ratio

The least numbers of those which have the same ratio with them are prime to one another.

G.277 numbers prime number

If two numbers be prime to one another, the number which measures the one of them is prime to the remaining number.

G.278 numbers prime number

If two numbers be prime to any number, their product also is prime to the same.

G.279 numbers prime product

If two numbers be prime to one another, the product of one of them into itself is prime to the remaining one.

G.280 numbers prime numbers

If two numbers be prime to two numbers, both to each, their products also is prime to one another.

G.281 numbers prime multiplying

If two numbers be prime to one another, and each by multiplying itself make a certain number, the products is prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter is prime to one another [and this is always the case with the extremes].

G.282 numbers prime sum

If two numbers be prime to one another, the sum is prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers is prime to one another.

G.284 prime number prime

Any prime number is prime to any number which it does not measure.

G.285 numbers multiplying make

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.

G.289 many numbers please

Given as many numbers as we please, to find the least of those which have the same ratio with them.

G.290 numbers find least

Given two numbers, to find the least number which they measure.

G.296 there many numbers

If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.

G.297 find numbers in continued proportion

Find numbers in continued proportion, as many as may be prescribed, and the least that are in a given ratio.

G.298 many numbers please

If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the extremes of them are prime to one another.

G.303 there many numbers

If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.

G.305 numbers there fall

If between two numbers there fall numbers in continued proportion with them, then, however many numbers fall between them in continued proportion, so many will also fall in continued proportion between the numbers which have the same ratio with the original numbers.

G.306 numbers prime numbers

If two numbers be prime to one another, and numbers fall between them in continued proportion, then, however many numbers fall between them in continued proportion, so many will also fall between each of them and an unit in continued proportion.

G.323 mean proportional numbers

If two mean proportional numbers fall between two numbers, the numbers are similar solid numbers.

G.342 many numbers please

If as many numbers as we please beginning from an unit be in continued proportion, by however many prime numbers the last is measured, the next to the unit is measured by the same.

G.345 numbers continued proportion

If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together is prime to the remaining number.

G.346 numbers prime number

If two numbers be prime to one another, the second will not be to any other number as the first is to the second.

G.347 there many numbers

If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the last will not be to any other number as the first to the second.

G.348 numbers investigate whether

Given two numbers, to investigate whether it is possible to find a third proportional to them.

G.349 numbers investigate possible

Given three numbers, to investigate when it is possible to find a fourth proportional to them.

G.362 odd number prime

If an odd number be prime to any number, it is prime to the double of it.

G.370 many numbers please

If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product is perfect.