# Checking Cases

Problems which look complicated can often be broken down into simplier scenarios, where we have more information. This makes the individual case easier to consider, which allows us to solve the entire problem.

How many triples of positive integers are there such that $a \times b + c = 8$?

A) 4
B) 8
C) 12
D) 16
E) 20

Solution: If $a = 1$, then we can have $b = 1$ to 7, which gives us 7 solutions.
If $a = 2$, then we can have $b = 1$ to 3, which gives us 3 solutions.
If $a = 3$, then we can have $b =1$ to 2, which gives us 2 solutions.
If $a =4$ to 7, we can only have $b = 1$. This gives us 4 solutions.
Hence, in total, there are $7 + 3 + 2 + 4 = 16$ solutions.

Note by Arron Kau
6 years, 11 months ago

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.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 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"
MathAppears 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} $\boxed{123}$