$$\\\small{\text{
d. What is the largest positive integer $n$ such that 1754, 1457 and 368 }}\\
\small{\text{
all leave the same remainder $r$ when divided by $n$?
}}$$
I.
$$\small{\text{
In a. we habe the difference $1754-1457 = 297 $
}} \\
\small{\text{
In b. we habe the difference $1754-368= 1386 $
}} \\
\small{\text{
In c. we habe the difference $1457 -368= 1089 $
}} \\
\begin{array}{lrcl}
\end{array}
}}$$
II.
$$\small{\text{
The prime factorisation all differences:
}} \\ \\
\small{\text{$
\begin{array}{lrcl}
(1) & 297 &=& 3^3\cdot 11 \\
(2) & 1386 &=& 2\cdot 3^2\cdot 7 \cdot 11 \\
(3) & 1089 &=& 3^2 \cdot 11^2 \\
\end{array}
$}} \\$$
III.
$$\small{\text{
The greatest $n$ is the greatest common divider of the differences:
}} \\ \\
\small{\text{$
\begin{array}{lrcll}
(1) & 297 &=& 3 &\mathbf{ \cdot 3^2 \cdot 11 }\\
(2) & 1386 &=& 2\cdot 7 &\mathbf{ \cdot 3^2 \cdot 11 }\\
(3) & 1089 &=& 11 &\mathbf{ \cdot 3^2 \cdot 11 }\\
\end{array}
$}} \\
\small{\text{$
\boxed{~~
n= gcd {(297,1386,1089)}=99 ~~}
$}}$$
IV.
$$\small{\text{
The greatest $n$ is $3^2\cdot 11 = 99$. The same remainder is 71 :
}} \\ \\
\small{\text{$
\begin{array}{lrcll}
(1) & 1754 &\equiv & 71 \pmod{99} \\
(2) & 1457 &\equiv & 71 \pmod{99} \\
(3) & 368 &\equiv & 71 \pmod{99}
\end{array}
$}} \\$$
