So i saw this question and im not sure how to do this without spending loads of time

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 terms is a perfect square. For example: 2, 6, 27 is a Shonk sequence since 6 > 2 and 27 > 6 and 2×6×27 = 324 = 182 .

(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?

c) Determine the length of the longest Shonk sequence, each of whose terms is an integer between 1 and 12, inclusive. This means that your solution should include an example of a sequence of this longest length, as well as justification as to why no longer sequence is possible.

(d) A sequence of four terms a, b, c, d is called a super-duper-Shonkolistic sequence (SDSS) exactly when each of a, b, c, d and a, b, c and b, c, d is a Shonk sequence. Determine the number of pairs (m, n) such that m, 1176, n, 48 400 is an SDSS.

Guest Apr 6, 2021

#1**0 **

*(a) If 12, x, 24 is a Shonk sequence, what is the value of x?*

I'll take this one cuz it's the easiest. We see that 12 • 24 is 288, which can be written as 2 • 144.

The 144 is a square so we need to find a number that when multiplied by the 2 becomes a square.

Note that 36 is a square and half of it lies between 12 and 24, so let's use **18**.

Check: 12 • 18 • 24 = 5184 .... and 5184 is a square, namely 72^{2}.

_{.}

Guest Apr 6, 2021