A finite sequence of numbers is called AWESOME if each subsequent member of the sequence, except the first one, is larger than the previous one and if the product of all the members of that sequence is a perfect square. For example, sequence 2,6,27 is an AWESOME sequence. Determine the natural numbers x and y such that the sequence 28,x,y,65 is an AWESOME sequence.
Is this a disordered sequence? Because I've tried so many times and I just can't see a pattern in this sequence. All I know is that x and y should be numbers between 28 and 65 and y is greater than x. Can please someone help me solve it?
$\endgroup$ 91 Answer
$\begingroup$hint
You look for $ x,y $ such that$$x<y<65$$and$$65\times 28xy=13\times 5xy\times 7\times 2^2$$is a perfect square. For example, you can take$$x=13 \text{ and } y=5\times 7=35$$
or$$x=35 \text{ and } y=13\times 2^2=52$$
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