Can you answer this immediately?

Number Theory Level 2

$\color{#20A900}5 + \color{#3D99F6} 4 \times \color{#20A900}5 + \color{#3D99F6}4 \times \color{#20A900}5^{ \color{#D61F06}2} + \color{#3D99F6}4 \times \color{#20A900} 5 ^{\color{#D61F06}3} + \color{#3D99F6}4 \times \color{#20A900}5 ^{ \color{#D61F06}4} = \ ?$

4 \times 5^ 5

3 \times 5^6

5^6

5^5

5 solutions

Uyen Nguyen
Jan 28, 2015

My way to solve this problem! Right???

That's a nice way of looking at it.

I was thinking of number bases when I created this problem, and how $44444_5 + 1_5 = 10000_5$ .

Chung Kevin - 6 years, 4 months ago

Joceleonard Tonido
Jan 29, 2015

5+4x5 + 4x5^2 + 4x5^3+4x5^4 = 5+20(1+5+25+125) =5+20(156) =5+624(5) =625(5) =(5^4)5 =5^5

Indigo Cat
Apr 9, 2015

This took me longer than "immediately" but it worked.

Factor out the 5.

5(1+4+4 * 5+4 * 5^2+4 * 5^3)

Add the 1 and 4, then factor again and again until you get 5^5.

Oh, that's another way to obtain the "base adding", by showing the addition step in each digit.

Chung Kevin - 6 years, 2 months ago

Cool! Would you mind explaining how this connects to the base solution?

Indigo Cat - 6 years, 2 months ago

i used this way too

Elsarnagawe Mohamed - 6 years ago

Abdo First
Jun 5, 2015

5 + 4*( (5^5-1)/(5-1) - 1 ) = 5^5

Since r^0 + r^1 + r^2 + ... + r^n = (r^(n+1))/(r-1)

Anurag Pandey
Jul 26, 2016

I did this way.

5 +(4 \times 5)+(4 \times 5^2)+(4 \times 5^3)+(4 \times 5^4)

=5 + [(4 \times 5)(1+5+5^2+5^3)]

Now using sum of a GP

=5 + [(4 \times 5) \frac {5^4 -1}{5-1} ]

=5 + [(4 \times 5) \frac {5^4 -1}{4} ]

=5 + [ 5 \times (5^4 -1)]

=5 + (5^5) - 5

=<\boxed{5^5}>

I did some mistake I think in using the latex could somebody help please. I am unable to use latex properly.

Anurag Pandey - 4 years, 10 months ago

Can you answer this immediately?

5 solutions

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