Thanks a lot, sir, for your work. I am seeing, how I become better everyday. This is the result of your work as well as mine
@gabiturbinca91022 жыл бұрын
If u use fermats theorem u get that 8^4k=1mod 5 from which 8^(4k+1)+47=8+47=0mod 5, and also 8^4k=2^12k=1mod 13,from which 8^(4k+3)+47=512+47=0mod 13 n even its pretty easy to show that 8^n+47=1+47=0 mod 3
@magicmeatball40132 жыл бұрын
Now prove fermats theorem
@霍金本人2 жыл бұрын
@@magicmeatball4013 Google is your best friend
@ignaciobenjamingarridoboba20712 жыл бұрын
@@magicmeatball4013 the induction proof is not hard
@zhangmike48522 жыл бұрын
the solution is amazingly simple!
@kobethebeefinmathworld9532 жыл бұрын
For Claim #2, I'll suggest using Fermat's little theorem to make the proof shorter and cleaner.
@mahmoudalbahar16412 жыл бұрын
Thank your for great videos... And I suggest that you make video about combinatorial proof for Hexagon identity in pascal triangle.
@goatgamer0012 жыл бұрын
if 8^n is 1 mod 3, then its pretty obvious. if it isnt, it is either divisible by 5 or 13.
@9core2 жыл бұрын
yeah this type of problem is almost always easily solvable using that method
@9core2 жыл бұрын
yeah this type of problem is almost always easily solvable using that method
@johnnath41372 жыл бұрын
If n is even let n = 2m. We have 8*n + 47 = 64^m + 47 ≡ 1^m + 47 (mod 3) ≡ 0 (mod 3) ⇒ 3|(8^n + 47) for even n. If n is odd, there are two cases (i) n = 4m + 1 or (ii) n = 4m + 3. Case (i): 8^n + 47 = (4)(256^m) + 47 ≡ (4)(1^m) + 47 (mod 17) ≡ 0 (mod 17) ⇒ 17|(8^n + 47) for odd n of form 4m +1 Case (II): 8^n + 47 = (64)(256)^m + 47 ≡ (1)(1^m) + 47 (mod 3) ≡ 0 (mod 3) ⇒ 3|(8^n + 47) for odd n of form 4n + 3. Thus 8^n + 47 is composite for all n.
@lgooch2 жыл бұрын
For odd case why didn’t we just do 2m+1?
@johnnath41372 жыл бұрын
@@lgooch Because there are two types of odd numbers, type 4m + 1 and type34m + 3, and "2n + 1" doesn't discriminate between them. And each type requires a different argument. I'll illustrate further by a nice example. Every prime of the form 4m + 1 can be expressed as the sum of two integer squares but no prime of the form 4m + 3 can be so expressed (a result due to Fermat).
@lgooch2 жыл бұрын
@@johnnath4137 ah thanks
@leif10752 жыл бұрын
@@johnnath4137 but 47 equals 2 mod 3 not zero mod 3?
@johnnath41372 жыл бұрын
@@leif1075 1 + 47 = 48 ≡ 0 (mod 3).
@richardfredlund38022 жыл бұрын
with the heavy hint re divisibility by 3,5,13 I twigged how to do this by about minute 7.
@bait66522 жыл бұрын
Interesint authoer typically likes to play with smaller modulo/negative...but calculated 64^2....instead of just using 64(1,-1,1,-1,1)for %3,5,13
@azai.mp42 жыл бұрын
Great example of how trying out small values can help you come up with a proof!
@pandagameryt62889 ай бұрын
we know that for a prime p>=7, p=1,-1 (mod 6). if 8^n +47 is a prime, then 8^n +47 = +1,-1 (mod 6). but 8^N+47 = 2^n -1 which should be =1,-1 (mod6). since -1 is is impossible, we take 1. so, 2^n= 2 (mod 6). 2^n-1 =1 mod6. which is never possible for n>=1and the given eq implies p>=55. so, 8^n +47 is never a prime.
@amagilly2 жыл бұрын
Another masterpiece.
@krishnanadityan20172 жыл бұрын
When n is even, the given number is divisible by 3.
@SQRTime2 жыл бұрын
Hi Krishnan. If you are interested in math competitions, please consider kzbin.info/www/bejne/r6C1dp-gq6x_pa8 and other videos in the Olympiad playlist. Hope you enjoy 😊
@paulchapman80232 жыл бұрын
Every power of 4 is two less than a multiple of 3, every power of 64 is also a power of 4, and 47 mod 3 = 2. Yup, that checks out.
@maelmao2 жыл бұрын
What’s the tablet you use ?
@snehasismaiti3422 жыл бұрын
I can't do this problem can anyone solve this "Find the smallest number which has 144 divisors 10 of which are consecutive number."
@kyooz47762 жыл бұрын
110880=2⁵×3²×5×7×11 notice that because you need to have 10 consecutive divisors at least one of them has to be divisible by 8=2³ and another one by 9=3². You then want to have the smallest primes possible in the factorization of this number. Because 144=2⁴×*3²* two terms will have exponent 3n-1. You can then either have 2³×3^(3-1)×5^(3-1)×7×11 which has 4×3×3×2×2=144 divisors or 2^(6-1)×3^(3-1)×5×7×11 which also has 6×3×2×2×2=144 divisors. The second one is smaller so the answer is 110880
@snehasismaiti3422 жыл бұрын
@@kyooz4776 why 3n-1 as exponent of 2 terms
@leif10752 жыл бұрын
@@kyooz4776 but the problem as stated doesn't make sense..144 divisors of 10..10 only has 4 divisors 1,2,5 and 10..and what does he mean consecutive numbers?
@kyooz47762 жыл бұрын
@@leif1075 read again
@kyooz47762 жыл бұрын
@@snehasismaiti342 because there are 144 divisors, search up number of divisors formula
@dimitardimitrakov28412 жыл бұрын
Proof by contradiciton without mods: 8^n + 64 - 17 = p where p is prime p>=55 and n>2 . Then 8^(n-2)+1 = (p+ 17)/64. left hand side is odd. In the right hand side p is odd because prime => p+17 is even . let p+17 = 2k. Then right hand side becomes k/32 and it must be odd because left hand side is odd. therefor k is odd and let k=2t+1. Finally we have 32*8^(n-2) + 32 = 2t+1 => something even = 2t - 31 which is odd.
@sirak_s_nt9 ай бұрын
You cannot claim that k is odd. Even though k/32 is odd, k can be 96 for which k/32 is odd. Use your common sense, if k is odd, how come 32 is going to reduce it to an integer? 32 cannot divide odd numbers.
@ReckonClasses2 жыл бұрын
Nice question
@ElmanMaharramov2 жыл бұрын
Where can you found this question? Please tell me it
@ibrahimagazade94182 жыл бұрын
😅😅
@ElmanMaharramov2 жыл бұрын
@@ibrahimagazade9418 tapdın burdada məni😅
@xeyalehemzeyeva082 жыл бұрын
😂
@euleri02 жыл бұрын
Indigenous!! ⭐
@Szynkaa2 жыл бұрын
tbh awful question, typical checking modulo prime numbers and praying to god for it to work without checking too many of them
@SQRTime2 жыл бұрын
Hi Mr Sz. If you are interested in math competitions, please consider kzbin.info/www/bejne/r6C1dp-gq6x_pa8 and other videos in the Olympiad playlist. Hope you enjoy 😊
@muhendisgenc82162 жыл бұрын
Woooow
@SQRTime2 жыл бұрын
Hi. If you are interested in math competitions, please consider kzbin.info/www/bejne/r6C1dp-gq6x_pa8 and other videos in the Olympiad playlist. Hope you enjoy 😊
@ElieBensaid4 ай бұрын
14 mins just to study that number in 3 different mods, kind of a waste of time
@aftorucova9980 Жыл бұрын
Does anyone have JBMO 2022 shortlist?
@wookong17232 жыл бұрын
loveeeeeeee
@amirhosseinmohajerinejad13932 жыл бұрын
Nice
@atpugnes2 жыл бұрын
I am little curious. Whenever you multiply, you start from the most significant digit. How is it so? Normally we would start from units place.
@sdspivey2 жыл бұрын
He isn't multiplying, he's just copying the result he already calculated.
@bornfromstardust15262 жыл бұрын
The method is called, Calculator Inclusion method.
@poproporpo2 жыл бұрын
Yes, even though he most likely copied computations that he had previously completed, there are mental math calculation methods that allow you to calculate from the most significant digit to the least significant digit. As you calculate, you would have to go back and change the result that you have already completed if you need to carry digits. Many mental arithmetic competitors prefer this method iirc.
@EternalLoveAnkh2 жыл бұрын
I don't mean to criticize but wouldn't it be easier to ask if it is ever prime? RJ
@Sp1tz1fy2 жыл бұрын
For photo math, it turns out as a exponential function
@Ming-pt6wt Жыл бұрын
photo math can't do number theory questions.
@王剛-m7n2 жыл бұрын
3.5.13 mod cover 2n,4n+1, 4n+3
@JinhaoPan-np7zyАй бұрын
写过
@xeyalehemzeyeva082 жыл бұрын
Would you please tell me where to find it?
@ElmanMaharramov2 жыл бұрын
Məndə soruşmuşam hələ cavab vermiyib
@xeyalehemzeyeva082 жыл бұрын
@@ElmanMaharramov deyəsən vermiyəcək..
@joaozin0032 жыл бұрын
n=-infinity
@hijeffhere2 жыл бұрын
What app are you using to write these? Thank you!
@browhat69352 жыл бұрын
Try latex codecogs
@typo6912 жыл бұрын
it's onenote, he wrote in a community post once
@hijeffhere2 жыл бұрын
Thank you, people!
@donaldcoombs792 жыл бұрын
donnkey kong mario cart
@haleshs662 жыл бұрын
13:06 How can 4096^k always have a remainder of 1 when divided by 13 ???
@crazycat15032 жыл бұрын
When you multiply numbers, their remainders also multiplies
@mcwulf252 жыл бұрын
(4095+1)*(4085+1)*.... where 13|4095. Multiply out the brackets and each term apart from the 1*1*1*... is divisible by 13.
@hhgygy2 жыл бұрын
@@mcwulf25 Thanks I also missed that step why 64k would always yield a remainder of 1
@leif10752 жыл бұрын
@@mcwulf25 where did you get 4095 from? And why out the extra 1 there?
@mcwulf252 жыл бұрын
@@leif1075 All I am doing is replacing 4096 with 4095+1. That 4085 should be 4095.
@iDovahkiin2 жыл бұрын
You mean: "Is 8^n+47 is always prime"? Idk if I'm wrong but aren't a prime number is a number that is only dividable by itself and 1? So 8+47=55 which is a prime number??
@ervinforever99532 жыл бұрын
The word **only** is the key. 55 is divisible with 5 and 11 as well.
@iDovahkiin2 жыл бұрын
@@ervinforever9953 how did i not realize 🤦
@leif10752 жыл бұрын
Isn't there a way tonsolve without using mod??
@adb0122 жыл бұрын
Yes. The same way. Just use division and remainder. For example, instead of saying "note that 64^k is 1 (mod 3)" say "note that if you divide 64^k by 3 you get a remainder of 1, because 63 is multiple of 3".
@adb0122 жыл бұрын
@@adityaekbote8498 ... Given that the question is asking if something is prime, and prime means that it is only divisible by itself and 1, and divisible means that the reminder of the division is zero, I doubt that it can be proved without using this concept. You can disguise it as "being in the time table of...", or doing repeated subtractions instead of dividing, or stuff like that. But at the end of the day, to show that something is NEVER prime you need to show that is ALWAYS divisible by something other than itself and 1.
@dimitardimitrakov28412 жыл бұрын
yes, i wrote it in the comments, if you use odd vs even.
@mr.kaiden71598 ай бұрын
If u don't know fermat's theorm U can use (a+b)^n =b^n MOD a we divide into four case for n=4k+i,for k=0,1,2,3,... and i=0,1,2,3 Now, we know that's number is odd ans 47=-1 MOD 3 47=2 MOD 5 47=-2 MOD 7 47=2 MOD 9 47= 3 MOD 11 47= -5 MOD 13 For case i=0,2 8^(4k+i)=(9-1)^(4k+i) Because i is even and 3|9 then 8^(4k+i)=1 MOD 3 For i=1 so 4k+1 is odd 8^(4k+1)=(10-2)^(4k+1)=-2 MOD 5 For i=3 so 4k +3 is odd 8^(4k+3)=(13-5)^(4k+3)=(-5)^(4k+3) MOD 13=(-5)((26-1)^(2k+1)) MOD 13=(-5)(-1) MOD 13=5 MOD 13 So 8^n+47 always can divided by 3,5,or 13
@yoav6132 жыл бұрын
Why are you negative? Ask is 8^n+47 always composite,and be positive!