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Solve the Diophantine Equation 7^(a -7) - 7^(b-7)=2400
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[sonu_singh_]

this is two step Q
7^a-7 --7^b-7=2400
7^4=2401
we can write as,
7^4 -7^0=2400
by camparing this to the equation we can write
7^11-4 --7^7-7
hence a=11 and b=7

[Stewyg]

Dont know why. But I understand more Mathematics with these youtubevideos than I did in school. Although we had a very good teacher. Good explained!

[montynorth]

Let a-7 = u.
Let b-7 = v.
Equation becomes 7^u - 7^v = 2400.
7^4 = 2401.
7^0 = 1.
2401-1 = 2400.
Thus u = 4 and v = 0.
Then a-7 = 4.
a = 11.
Then b-7 = 0.
b = 7.

[marcod]

Hi!
What about this one: the given equation implies
7^(a-7) > 7^(b-7) so a-7>b-7 and a>b.

Let’s factor 7^(b-7):
7^(b-7)[7^(a-b) - 1] = 2400

Since 7 divides 7^(a-b) but not 1, it does not divide their difference.
So 7^(b-7) is the highest power of 7 that divides 2400. But 7 does not divide 2400, so b-7=0 which means b=7.

Then 7^(a-7) -1 = 2400, which leads to a-7=4 and so a=11. Done.

[broytingaravsol]

if purposely finding the multiples over 2400 which is based on 7, 2400=2401-1, then leading to each value of a and b

[acnmes]

7^(a-7)-7^(b-7)=2400
((7^a)/(7^7))-((7^b/(7^7)) =2401-1
(7^a)-(7^b)=((7^4)-1)*(7^7)
(7^a)-(7^b)=(7^11)-(7^7)
a=11
b=7
Un peu moins rigoureux !!!
mais réponds à la question

[nirupamasingh]

Vvv nice presentation in v clear v easy steps

[זאבגלברד]

If you write down six numbers: 7^0, 7^1, 7^2, 7^3, 7^4, 7^5 you see that only 7^4 - 7^0 = 2400

[manojitmaity]

This was the trickiest one ever solved by you...

[mattsemere]

How come do you have to include k when you were doing the substitution?

[marcovargasglobant]

Easy: 2400 = 2401 - 1 = 7⁴ - 7⁰
so: (a - 7)=4, (b - 7) = 0
therefore: a = 11 ; b = 7
greetings!

[billcame]

That's what I got. Did it in my head.

[johnbrennan]

Thanks tutor for another nice problem. I did it the exact same way because I remembered a similar problem you solved some time back using the same method.

[alster]

Nice!!!! You made it look easy for me.

[mathsplus]

Very nice presentation, very clear steps. Thank you sir

[michaelpurtell]

By inspection can’t one just deduce that b=7 and c=11 since 7^4= 2401

[eckhardfriauf]

If one knows that 7^4 = 49 * 49 = 2401, then it is quite obvious that a and b are 11 and 7, respectively. NB: 50 * 50 = 2500, 2500 - 50 = 2450, 2450 - 49 = 2401.
Anyway, great riddle.

[PeterLE]

And he isn't telling that this is just one solution of an unlimited amount of solutions