That's Plenty of Digits There!

If $a= 10^{624}$ and $b=25$ , then

$10^x = (a + b)^2 - (a - b)^2 = a^2 + 2ab + b^2 - (a^2 - 2ab + b^2) = 4ab$

Because $b = 25$ , then $4b = 100$ . Moreover, since $a = 10^{624}$ , then $4ab = 100a = 10^{626}$ , giving us our final answer of 626.

Let $a=(10^{624}+25)$ and $b=(10^{624}-25)$

Thus by $a^2-b^2=(a+b)(a-b)$ we get, $10^x=(10^{624}+25+10^{624}-25)(10^{624}+25-10^{624}+25)$

$10^x=(2×10^{624})(2×25)$

$10^x=4×25×10^{624}$

$10^x=100×10^{624}$

$10^x=10^2×10^{624}$

$10^x=10^{2+624}$

$10^x=10^{626}$

$\boxed{x=626}$

$\begin{aligned} 10^x & = \left(10^{624} + 25 \right)^2 - \left(10^{624} - 25 \right)^2 \\ & = 10^{1248} +(50)10^{624} + 625 - 10^{1248} +(50)10^{624} - 625 \\ & = (100)10^{624} \\ & = 10^{626} \\ \\ \Rightarrow x & = \boxed{626} \end{aligned}$

$10^x=(10^{624}+25)^2-(10^{624}-25)^2$

$10^x=(10^{624}+25+10^{624}-25)(10^{624}+25-10^{624}+25$

$10^x=2(10^{624})(50)$

$10^x=100(10^{624})$

$10^x=10^{626}$

$\boxed{x=626}$

$Let\quad y = 10^{624} \\ i \rightarrow \quad (y+25)^2 = y^2 + 50y + 625 \\ ii \rightarrow \quad (y-25)^2 = y^2 - 50y + 625 \\ i - ii = 100y = 1000^{624} = 10^{626} \\Therefore \quad \boxed{x = 626}$

If we take x=10^624 and y=25, then we have:

(x + y)^2 - (x - y)^2

= (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)

= 4xy = 100x

= x^626

• I saw two different ways to solve this one, feel free to use the one you think it's easier.
• Method I : let's set $a = (10^{624} + 25)$ and $b = (10^{624} - 25)$ . $10^{x} = a^{2} - b^{2} = (a + b)(a - b)$ . $10^{x} = (2\times100^{624})(50) = (100\times 10^{624})$ .
• Since $100 = 10^{2}$ , we have $10^{x} = 10^{624 + 2} => x = 626$ .

• Method II : now we set $a = 10^{624}$ and $b = 25$ .
  $10^{x} = (a + b)^{2} - (a - b)^2$ .
  $10^{x} = a^{2} + 2ab + b^{2} - a^{2} +2ab - b^{2} = 4ab$ .
  $4ab = 4\times 25 \times 10^{624} = 100\times 10^{624}$ .
  $10^{x} = 10^{624 + 2}$
  
  
  
  

• So, we conclude x = 626 .

Python:

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def left(x):
    return 10**x
def right(x):
    return (10**624+25)**2-(10**624-25)**2
x = 0
while left(x) != right(x):
    x += 1
print "Answer:", x

Note: I kind of cheated by looking at the input field to see that it wasn't asking for a decimal answer, so I knew the answer was an integer.

$10^x=(10^{624}+25)^2-(10^{624}-25)^2=((10^{624}+25)-(10^{624}-25))((10^{625}+25)+(10^{625}-25))=(10^{624}+25-10^{624}+25)(10^{625}+25+10^{625}-25)$

$=2 \times 25 \times 2 \times 10^{624}=100 \times 10^{624}=10^2 \times 10^{624}=10^{2+624}=10^{626} \implies x=\boxed{\large{626}}$

lets say 10^624 +25= a, and 10^624-25=b , then the equation is a^2-b^2= (a+b)(a-b), which substituting gives us that 10^x=2(10^624)(50). so 10^x=100(10^624), so x=626

The Only terms will remain are $(2\times 1st\space term\times 2nd\space term)$ for each

So it will be;

$10^x=4\times25\times10^{624}=100\times10^{624}=10^2\times10^{624}=10^{626}$

$x=626$

Using the algebraic identity, a^{2} - b^{2} = (a-b)(a+b), we get $10^{x} = (10^{624}+25-10^{624}+25)(10^{624}+25+10^{624}-25) =(50)(10^{624})(2) (10^{x})/(10^{624})=100 10^{x-624}=10^{2} x-624=2 x=625$

If ä=10^624 & b=5 then 10^x= ( ä+b )^2 - ( ä-b )^2 = ä^2+2ab+b^2 -( ä^2-2ab +b^2) =4ab =4 10^624 25 =10^2*10^624 =10^626 » x = 626

10^x = (10^624 + 25 + 10^624 - 25)(10^624 + 25 - 10^624 + 25) =

(2 . 10^624 )(50) =10^626

x = 626

Write equation as:

(A+B)^2 - (A-B)^2 =10^x where A=10^624 and B=5^2 (or [10/2]^2)

expanding left side gives:

4AB=10^x

= 4 x 10^624 x [10/2]^2 = 10^x

= [4/4] x 10^624 x 10^2 = 10^ x

=10^624 x 10^2 = 10^x

therefore:

x = 624 + 2 = 626

Let y=10^(624) +25 Note ^ means power to Let Z=10^(624) -25 Then Y^2-Z^2 =(Y+Z)(Y-Z)=100*10^624 =10^626 Then X=626