Number Theory

(a)Use the Euclidean algorithm to find $\displaystyle d$, the greatest common divisor of 703 and 399.

(b)Find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle d=703a+399b.$

(c)Hence find integers $\displaystyle p$ and $\displaystyle q$ such that $\displaystyle 76=703p+399q.$

Use the Euclidean algorithm to find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 1595a + 1218b = 29.$

Use the Euclidean algorithm to find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 306a + 119b = 17.$

Use the Euclidean algorithm to find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 634a+87b=1.$

Use the Euclidean algorithm to find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 533a + 455b = 13.$

Express $\displaystyle 231_{11}$ in base 7.

Express $\displaystyle 572_{10}$ in base 9.

(a)Use the Euclidean algorithm to find $\displaystyle d$, the greatest common divisor of 1118 and 416.

(b)Hence find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 1118a + 416b = d.$

(a)Use the Euclidean algorithm to find $\displaystyle d$, the greatest common divisor of 1428 and 567.

(b)Find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 1428a+567b=d.$

Express $\displaystyle 3442_{5}$ in base 9.