Question

Find the greatest common divisors. You should be able to do parts (a)-(c) by hand, but technology is OK for the rest.

(a) $\left(56,72\right)$ (b) $\left(24.138\right)$ (c)$\left(112,57\right)$

(d) $\left(143,231\right)$(e) $\left(306,657\right)$(f) $\left(272,1479\right)$

(g) $\left(4144,7696\right)$ (h) $\left(12378,3054\right)$

Short answer

Thus the required answer is:

$$\begin{gathered} \left(a\right)\left(56,72\right)=8\left(b\right)\left(24,138\right)=6\left(c\right)\left(112,57\right)=1 \\ \left(d\right)\left(143,231\right)=1\left(e\right)\left(306,657\right)=9\left(f\right)\left(272,1479\right)=17 \\ \left(g\right)\left(4144,7696\right)=592\left(h\right)\left(12378,3054\right)=6 \end{gathered}$$

Step-by-step solution

Greatest Common Divisor

The Greatest Common Divisor of two integers $a$ and $b$ is defined as the largest integer $d$ that divides both $a$ and $b$.

Find Greatest Common Divisor

The divisors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56 and the divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Therefore, the common divisors are 1, 2, 4, and 8. This implies that the greatest common divisor is 8, that is,$\left(56,72\right)=8$ .

Find Greatest Common Divisor

The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24, and the divisors of 138 are 1, 2, 3, 6, 23, 46, 69, and 138.

Therefore, the common divisors are 1, 2, 3, and 6. This implies that the greatest common divisor is 8, that is, $\left(24,138\right)=6$.

Find Greatest Common Divisor

The divisors of 112 are 1, 2, 4, 7, 8, 14, 16, 28, 56, and 112, and the divisors of 57 are 1, 3, 19, and 57.

Therefore, the common divisor is 1. This implies that the greatest common divisor is 1, that is, $\left(112,57\right)=1$.

Find Greatest Common Divisor

Use the TI calculator to find the result. Follow the step to determine the GCD.

• Enter $a=143$ and $b=231$ in the TI calculator.
• Press Enter to obtain GCD.

The result is shown as:

$\begin{array}{rcl}AU+BV& =& GCD\\ & =& 11\end{array}$

Thus, $gcd\left(143,231\right)=1$.

Find Greatest Common Divisor

Use the TI calculator to find the result. Follow the step to determine the GCD.

• Enter $a=143$ and $b=231$ in the TI calculator.
• Press Enter to obtain GCD.

The result is shown as:

$\begin{array}{rcl}AU+BV& =& GCD\\ & =& 9\end{array}$

Thus, $gcd\left(306,657\right)=9$.

Find Greatest Common Divisor

Use the TI calculator to find the result. Follow the step to determine the GCD.

• Enter $a=272$ and $b=1479$ in the TI calculator.
• Press Enter to obtain GCD.

The result is shown as:

$\begin{array}{rcl}AU+BV& =& GCD\\ & =& 17\end{array}$

Thus, $gcd\left(272,1479\right)=17$

Find Greatest Common Divisor

Use the TI calculator to find the result. Follow the step to determine the GCD.

• Enter $a=4144$ and $b=7696$ in the TI calculator.
• Press Enter to obtain GCD.

The result is shown as:

$\begin{array}{rcl}AU+BV& =& GCD\\ & =& 592\end{array}$

Thus, $gcd\left(4144,7696\right)=562$.

Find Greatest Common Divisor

Use the TI calculator to find the result. Follow the step to determine the GCD.

• Enter $a=12378$ and $b=3054$ in the TI calculator.
• Press Enter to obtain GCD.

The result is shown as:

$\begin{array}{rcl}AU+BV& =& GCD\\ & =& 6\end{array}$

Thus, $gcd\left(12378,3054\right)=6$.