Tricky algebra

The sum 7/12 + 9360/d is an integer. Compute the smallest positive integral value for d.

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

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

Write the second fraction as c.m/c.n where the greatest common divisor of m and n is 1. If 7/12 + m/n is to be some integer k, then 7n + 12m = 12nk ->12m = 12nk - 7n -> m = nk - 7 (n/12) . Therefore 12 divides n. We also have that 7n = 12nk - 12m -> 7 = 12k - 12m/n . But since m and n are relatively prime, we have that n divides 12. So n = 12. Therefore our goal is to have that when the fraction is reduced the denominator is 12. We need to find the smallest c so that the numerator 9360 has no factors of 12. Dividing 9360 by 12 two times shows that 9360 = 12^2 * 65. So c = 12^2 for the smallest value of d, and d = 12^2 12 = 1728. We must check that this is indeed an integer, and it is because 7/12 + 65/12 = 72/12 = 6. So d = 1728. CMIIW

snah a - 8 years ago

I think the answer is 1728 and the integer is 6

Avraam Aneleitos - 8 years ago

Sumit Sarkar - 8 years ago

you should specify that d is positive

Justin Wong - 8 years ago

it was specified so.

Jonathan Wong - 8 years 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}`	$\boxed{123}$