Show that any non-negative rational integer may be written as the sum of two squares if it is of the form.

Step-by-step explanation:

first of all a rather trivial point: a square is a sum of two squares (of which usually one will need to be 0²)

now you need a lemma: if a is a sum of two squares and b is a sum of two squares, then the product ab is also a sum of two squares.

this is easy to see in terms of the multiplication of two Gaussian integers:

(a²+b²)(c²+d²) = |a+ib|² |c+id|² =|(a+ib)(c+id)|² = |(ac−bd) + i(ad+bc)|² = (ac−bd)² + (ad+bc)²

since 2=1²+1² the lemma implies any power of two is a sum of two squares.

a well-known theorem tells us that any prime of the form 4n+1 is the sum of two squares, so the lemma implies that any product of such primes p1^(k1) p2^(k2)⋯pm^ (km) is a sum of two squares.

finally to multiply a sum of two squares by a square (of the form q2j11q2j22⋯q2jnn) is again a sum of two squares.

the proof of the theorem about primes of the form 4n+1 is a fairly easy consequence of two basic results: Wilson's theorem and the fact that the Gaussian integers are a Euclidean ring and therefore are a Unique Factorization Domain. Wilson's theorem is a simple exercise in modular arithmetic. To demonstrate a division algorithm for the Gaussian integers is a little more technical but of an elementary nature.

answer: the value should be used for i in the formula of calculation the monthly payment , the interest rate per period is 0.85.

step-by-step explanation:

the given rate of interest i= 85%

in calculating the monthly payment for a loan with an interest rate compounded monthly, the value of the interest rate per period we use in the formula must be in decimal form.

to convert it in decimal, we divide it by 100, we get

hence, the value should be used for i in the formula of calculation the monthly payment , the interest rate per period is 0.85.