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Hello all, I have a question in response to the an e-maths question given below:
The product of two consecutive positive even numbers is 60024. What is the smaller of these numbers?
P.S: I do know the workings to this question, but I have a doubt about the workings.
The workings:
Let the smaller no. be p. The other no. be p+2.
And so on....
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Why must they use p+2? Can't they use other alternatives like p+1, p+3 etc? Please explain. Thanks so much. :)
If p is even, then p+1 and p+3 are odd.Quote:
Originally Posted by Yunie
Quote:
Originally Posted by krtxmrtz
So that means I could also use other even alternatives for p like p+4 or p+6 right? So long as it is an even number right? And that also means that, if the question states that p is odd, then I must use odd numbers for the value of p like p+1 or p+3 right?
Thanks again. :)
Right. As long as they are consecutive, i.e., their difference is 2.Quote:
Originally Posted by Yunie
Yes, if one number is odd and the other is even you can use p and p+1, or p+8 and p+9,...Quote:
Originally Posted by Yunie
If both are odd, then again you must use p and p+2, or p-34 and p-32, etc.
Don't forget they are consecutive numbers, so there are no alternatives to using p + (p + 2) = 60024
If p is odd you still have to use p + (p+2): 3 + 5 for example, or 15 + 17, or 31 + 33 (they are consectutive!)
If you wanted to find the larger number, you would use p + (p - 2)...
This has probably already been beaten to death, but I wanted to put in my 2 cents.
Your choice of "p" doesn't honestly matter--you could call "p" anything you feel like, and define it however you wish. It could be the square root of the inverse cosecant of the smaller of the two numbers. You simply have to connect the smaller and larger of the two numbers algebraically to 60024.
For example, let p = sqrt(smaller number), then p^2 = (smaller number), and p^2 + 2 = (larger number). Thus, 60024 = smaller number * larger number = (p^2 * [p^2 + 2]), or p^4 + 2p^2 = 60024. You could solve for p, and use it to find the smaller number, but... it's stupid.... You could similarly say p = sqrt(arccsc(smaller number)) and solve for the smaller number through p. Again, it doesn't really matter. No matter what you do, you'll find that the smaller of the two numbers is uniquely identified [the functions you use must actually be defined properly in the reals... but that's about it].
Edit: Oh, and yup, they can use p+1, p+3, so long as it's understood that p+1 is the smaller of the two numbers and p+3 is the larger of the two numbers. Then (p+1)(p+3) = 60024, p^2+4p+3 = 60024 -> p = 243, -247 [the second root gives a negative smaller number, and the number is supposed to be positive, so it's thrown out]. Thus, the smaller number is p+1, or 244, and the larger is p+3, or 246. 244*246 = 60024, as required.