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Name: Find integers q and r such that 250=16q+r and 0≤r<16 . Explain careful
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Description: Find integers q and r such that 250=16q+r and 0≤r<16 . Explain carefully why gcd(250,16)=gcd(16,r). Check that this equation is in fact true by explicitely finding all the factors of 250, 16, and r. (One possible way to list the factors is to first f

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Find integers q and r such that 250=16q+r and 0≤r<16 . Explain carefully why gcd(250,16)=gcd(16,r). Check that this equation is in fact true by explicitely finding all the factors of 250, 16, and r. (One possible way to list the factors is to first find the prime factorization of each number.)

Expert Solution

Step 1

$250 = 16 q + r$

where q is the quotient and 'r' is the remainder

Lets apply Euclid division algorithm

Dividend = divisor * quotient + remainder

we know that 15 times 16 = 240

So ,

$250 = 16 \cdot 15 + 10 250 = 240 + 10 q u o t i e n t q = 15 r e m a i n d e r r = 10$

As per Euclid division algorithm

divisor=16, and remainder r=10

Both 16 and 10 are divisible by 2

So gcd of (250,16) is also 2

gcd(250,16)=gcd(16,10)

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