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Math Contest Question
aly0303
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A month ago
Math Contest Question
A
Shonk sequence
is a sequence of positive integers in which
each term after the first is greater than the previous term, and
the product of all the terms is a perfect square
For example: \(2, 6, 27\) is a Shonk sequence since \(6>2\) and \(27>6\) and \(2\cdot 6\cdot 27=324\) or \(18^2\)
a. If 12, x, 24 is a Shonk sequence, what is the value of x?
b. If 28, y, z, 65 is a Shonk sequence, what are the values of y and z?
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habiba
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Quote from: aly0303 (A month ago)
a. If 12, x, 24 is a Shonk sequence, what is the value of x?
12 * x * 24 will multiply to a perfect square.
This means that when you square root this perfect square, it outputs an integer.
Let's call this perfect square y for reference:
12*x*24 = y
Theoretically, both sides of this equation can be square rooted and it'd result in an integer on both sides. So, let's square root both sides:
sqrt (12*x*24) = sqrt(y) [ignore the right side for now on focus on the left]
distribute the square root to each factor:
√12 √x √24
reduce the radicals:
(2 √3) * √x * (2 √6)
combine √3 and √6 to get √18. The √18 is 3 √2. This leads us to the following expression
2 * 2 * 3 * √2 * √x
12 * √2 * √x
Combine √2 with √x since they're both square roots:
12 * √ (2*x)
Now you need to find a number that when multiplied to 2 gives you a perfect square. Remember, this number needs to be between [12 and 24], so choose wisely. The number that fits is x = 18
When x = 18, we get √36, and since 36 is a perfect square, this makes the left side:
12 * 6 = 72
So the √y = 72
y = 72^2 = 5184
You can do (b) the same way!
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b. If 28, y, z, 65 is a Shonk sequence, what are the values of y and z?
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bio_man
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bio_man
wrote...
#2
Educator
A month ago
Edited:
A month ago, bio_man
This is some next level math!
Using the same approach at @habiba, we get:
Notice that I strategically chose numbers between the boundary that were divisible by 13, 5, and 7. I needed to do this so that they could become perfect squares.
Any further questions, let us know.
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aly0303
wrote...
#3
A month ago
That's amazing! Thank you both!!!
Can you help me with part (c)
Quote
c. Determine the length of the longest Shonk sequence, each of whose terms is an integer between 1 and 12, inclusive. Your solution should include an example of this longest length, as well as justification as to why no longer sequence is possible.
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bio_man
wrote...
#4
Educator
A month ago
Start with 12!
Write its prime factorization.
Square root each of them to see if they're perfect squares. You know it's a perfect square if its exponent is even.
Go from there
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