Olympiad Maths Problem : Can you solve for Exponents X and Y
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- Опубліковано 14 жов 2024
- Join me in this new video on how to solve a challenging olympiad type maths problem. Can you do it?
The problem is find x and y from 2^x-2^y=496
#exponentialequation #exponential #olympiadmathematics
#mathsplus
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Oh yes! Awesome question!!! Thank you so much.
P.S. TOO EASY BRO
Thank you. Have a great day.
@@mathsplus01 You are most welcome and thank you for your kind words and have a good day
Welcome everyone to this Olympiad Maths Question. If anyone has any questions about it I will be delighted to answer you below🤔
In old days 1st you try to guess and then you solve. This one is easy to guess.
Take the closest power of 2 that is higher than 496 and it is 512 and the difference is power of 2 as well. Problem solved in 2 seconds.
Very interesting problem. Thank you for your videos!
Cheers onenote. Thanks for the support and bless you
Motivation: when you have a difference on one side and a number on the other, you always try to convert the difference into a product and then you can compare the product with the factors of the number. We know that 2^n > 0 so it follows that 2^x > 2^y, i.e. x > y. So we take the smaller factor 2^y out of both of the terms on the LHS and factorise the RHS:
2^y * (2^(x-y) - 1) = 2^4 * 31. The only way that can be true (for integer x and y) is if 2^y = 2^4, and 2^(x-y) - 1 = 31, meaning 2^(x-y) = 32, so 2^(x-y) = 2^5. That gives y = 4 and x = 4+5 = 9.
Thank you, grand solution.
2^x-2^y can be used to find perfect numbers where x=1, 3, 5, 7... and y=0, 1, 2, 3... Not all numbers from this formula are perfect but all perfect numbers fit this formula.
Thank you Richard, very interesting
They make these things more complicated than they have to. Yes 2^x > 496 and 2^x > 2^y. Remember the powers of 2: 2,4,8,16, etc. The first one that is greater than 496 is 512 which equal to 2^9. 512 - 496 = 16 = 2^4. Therefore X = 9, and Y =4
I will support you bro
Now it is 397 subs it will raise in few days wait for that
Thank you. Blessings to you and your family.❤
@@mathsplus01 i did it
I had raisen your subscribers
Keep that confident and keep growing
Best of luck my dear friend
@@tatayyanaidugaraga801 I am very grateful for all your help. You have great spirit of kindness within you. Bless you. A million Thanks for you
@@mathsplus01 Now iam quiting
Good bye my dear friend
Good solution. Impressive as it appears simple in the end.
Always a big thanks for your support. Bless you from Ireland!
496 = 16*31 = 16* ( 16+8+4+2+1)
= 16*(32-1) = 2^9 - 2^4
Hereby x = 9 and y = 4
Excellent work
Let's take
x = log2(p) and y = log2(q)
The equation will becomes:
p-q = 496 ........ (Eq1)
Because of log2 we also have following constraints:
p>0 ...... (Eq2)
q>0 ...... Eq3)
Any tuple (p,q) satisfies Eq 1 2 3 is the solution. There are infinite solution.
Video should really state that we are finding integer values of x and y.
I clicked on this thumbnail without knowing what it was and then I was like “hey that sounds kind of like Maths Plus!” 😂😂😂 Nice video!
Haha Thank you Owls School of Maths Sir , glad to know the videos still circulate like that also.
@@mathsplus01 right!!! The algorithm is still serving it up! 🤣
Nice! Love these types of problems.
They are very popular also. I keep them simple and keep to real solutions so most people can understand.
did you try using screen record and writing on a digital whiteboard? All you need a cheap writing tablet. I would be able to give you a few tips if you would to do your videos like i do. But the way you do them on video works just fine also
@@mathsplus01 yes! So if you happen to scroll back in my history all my old videos were on the digital whiteboard and I actually kind of like it! But currently my digital setup is broken and I haven't gotten around to fixing it. I am not sure if i would go 100% digital again but would like to actually try and compare. So yes, when I give that a try again I could probably use some tips because I was actually struggling quite a bit with the setup! 🤣
It's kind of ironic because I created my website obviously but don't know how to work the whiteboard 🤣
@@owlsmath ok i understand, i think your maths is very good, the main way i improved in recent weeks was to do the video in sections, then merge together in adobe merge video for free. There is less pressure do all in one go also.
496 = 31 * 16 = 31 * 2^4 , then 2^(x-4) - 2^(y-4) = 31 , 31 = 32 - 1 --> 31 = 2^5 - 2^0 , then x-4 = 5 -> x=9, and similarly, y - 4 = 0 --> y =4
Thank you so much. Really good solution there👍
In one step 496=512_16 =2^9_2^4 then x=9, y=4
Thank you , excellent reply.
Interesting problem. I had a really different approach.
let’s represent 496 in binary
2^x-2^y=2^8+2^7+2^6+2^5+2^4
2^x=2^8+2^7+2^6+2^5+2^4+2^y
It’s easy to see that this is only possible when y = 4, because each term will get added to the one on its left eventually leading to one term remaining.
2^x=2^9
x=9
Therefore x=9, y=4
Thank you so much
You are most welcome❤ Have a nice day!
Yes enjoyed the whole video. It's class A work
Thank you Adam Mohamed, I am glad you liked it. Blessings from Ireland
I have been doing a chapter for the last few days... I thought I'll complete it today... 2 questions were left and I was just finding no interest in it for now... I came through your video... I found it really interesting, so I clicked and loved the problem... Great explanation.... Thank you so much 🙏
Always do the maths in small steps. This is how you make progress. I know you will succeed!
X=9,y=4
Good work Barry. Awesome!
X =9&y =4
Fantastic work, correct. Keep going and well done to you!
X=9; Y= 4
Well done!
x=9 dan y=4
Yes, correct. You really get them right. Well done!
Just memorizing my powers of 2 was enough
Still solved it so you did it. Well done!
x=k+y 2^k+y - 2^y = 496 2^y(2^k-1) = 496
2^y=16=2^4 y=4 2^k-1=31 2^k=32=2^5 k=5 x=5+4=9
( x=9 , y=4 )
512 - 16 = 496
512 = 2⁹ and 16 = 2⁴
=> x = 9 and y = 4
takes less than 30 seconds
Aren't you a genius
@@mathsplus01 Sorry, but to solve this problem in 30 seconds doesn't take a genius, if you have seen a similar problem on another channel. 😃
x=9 dan y=4