# Math desu yo, Math desu yo!

October 17th, 2009 by ben

I felt like it.

Here’s the design I’m sponsoring for the 2009-10 Arcadia Math Team t-shirt.

$i^2$

keepin’ it real

Anyways, let’s go with a really easy problem this time, okay? Great!

Um, I actually have more trouble coming up with “easy” problems, compared to hard ones. Okay, I’ll go with this one; it’s not too hard.

A number x is 56 less than a perfect square, and 124 less than another perfect square. What is x?

from October 2009 ARML Practice

### Related Posts:

1. k says:

… at shirt.

2. Benji says:

What about the shirt? I know, genius, huh?

At least it’s better than the Pikachu shirts from last year.

3. Andy says:

[spoiler]200[/spoiler]

… Easy!! ^_^

4. Benji says:

NOW PROVE IT! >:]

5. k says:

True.

6. Benji says:

Hold on, lemme enable LaTeX in comments.

7. Andy says:

Proof:
[spoiler]
200+56 = 256 = 16 squared
200+124 = 324 = 18 squared
[/spoiler]

8. Benji says:

Hank actually WANTED one of the Pikachu shirts.

9. Benji says:

Solution under the spoiler tag.

[spoiler]

Let x be the number we are looking for. From the problem:

\$latex x + 56 = m^2 \$

and

\$latex x + 124 = n^2 \$

Subtract the two equations from each other, and you get:

\$latex 124 – 56 = n^2 – m^2 \$

Factor the difference in squares.

\$latex 68 = (n+m)(n-m) \$

What are two divisors of 68 that can satisfy this? They are 2 and 34.

\$latex n+m = 34 \$

and

\$latex n-m = 2 \$

Solving this trivial system, we get n and m to be 16 and 18. Thus, x is 256 minus 56, or 200.

[/spoiler]

10. Andy says:

^_^ How to Solve ^_^
——————————-
[spoiler]

Uses a sequence of squares:

A = 1, 4, 9, 16….
The interval between each term is based off another series:
B = 3, 5, 7, 9
With the formula
b of n = 2n+1

b of n is the interval to the next square.

—————–
We are given that x is 56 from one square and 124 from another….
sooo…..

The distance between the two squares is:
124-56 = 68
Finding it as a sum of consecutive nums:
68 = 35+33
so there are 2 intervals between the squares…

33 is the first interval after the first square

From above
b of n = 2n +1
Solve!
2n = 32
n = 16

Sooo…..
16 is the first number to be squared: 256
x + 56 = 256
x = 200

[/spoiler]

Yay!!!
Life is good.

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