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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

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.4: Prime Factors And Greatest Common Divisor

Problem 33E: Use the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such that...

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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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