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Question 1. Use the Euclidean Algorithm to find the greatest common divisor of

by admin|Published May 18, 2026

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Question 1. Use the Euclidean Algorithm to find the greatest common divisor of

$\gcd(875,4075)$

Theorem 3. The Euclidean Algorithm. If $a$ and $b$ are positive integers, $b\neq0,$ and

$a=bq+r,\quad 0\leq r<b$

$b=rq_1+r_1,\quad 0\leq r_1<r$

$r=r_1q_2+r_2,\quad 0\leq r_2<r_1$

$\cdots$

$r_k=r_{k+1}q_{k+2}+r_{k+2},\quad 0\leq r_{k+2}<r_{k+1}$

then for large enough $k,$ say $k=t,$ we have

$r_{t-1}=r_tq_{t+1}$

and $(a,b)=r_t$

Let

$a=4075,\quad b=875$

Then,

$4075=(875)(4)+(4075-4(875))$

$=4(875)+575$

$\Longrightarrow a=4075,\quad b=875,\quad q=4,\quad r=575$

Then,

$875=(575)(1)+(875-(1)575)$

$=(575)(1)+300$

$\Longrightarrow b=875,\quad r=575,\quad q_1=1,\quad r_1=300$

Then,

$575=(300)(1)+(575-(1)(300))$

$=(300)(1)+275$

$\Longrightarrow r=575,\quad r_1=300,\quad q_2=1,\quad r_2=275$

Then,

$300=275(1)+(300-(1)(275))$

$=275(1)+25$

$\Longrightarrow r_1=300,\quad r_2=275,\quad q_3=1,\quad r_3=25$

Then,

$300=25(12)+0$

By The Euclidean Algorithm,

$r_3=r_t=25$

Thus,

$\gcd(875,4075)=25$

For Q2

$4075=(875)(4)+575$

$875=(575)(1)+300$

$575=(300)(1)+275$

$300=275(1)+25$

$300=25(12)+0$

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