find all positive integer solutions for $a^{3}bc = 24$

Find all positive integer solutions for $a^{3}bc = 24$ .

#Algebra

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$24 = 2^3 \cdot 3 = a^3bc$

Since the only prime factor whose exponent is at least 3 is 2, $a$ has to be 2. Then, there is a remaining factor of 3, but it can't be split up, so either $b$ or $c$ is 3 and the respective other is 1.

$(a,b,c) = (2,1,3)$

$(a,b,c) = (2,3,1)$

Now, there is also the possibility of $a = 1$ . In this case, $b$ has to be one of 24's divisors and $c$ is determined by that.

$(a,b,c) = (1,1,24)$

$(a,b,c) = (1,2,12)$

$(a,b,c) = (1,3,8)$

$(a,b,c) = (1,4,6)$

$(a,b,c) = (1,6,4)$

$(a,b,c) = (1,8,3)$

$(a,b,c) = (1,12,2)$

$(a,b,c) = (1,24,1)$

Henry U - 2 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

find all positive integer solutions for a3bc=24a^{3}bc = 24a3bc=24

Comments

find all positive integer solutions for $a^{3}bc = 24$