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# Number Theory Warmups

If numbers aren't beautiful, we don't know what is. Dive into this fun collection to play with numbers like never before, and start unlocking the connections that are the foundation of Number Theory.

\[ 2^{1}-2^{2}+2^{3}-2^{4}+2^{5}-2^{6}+2^{7} -\ldots +2^{2013}\]

Let \(a\) be the value of the expression above. Find the last two digits of \(a.\)

Given that in the 8-digit number \(\overline{\mathrm{ABCDEFGH}}\),

(i) \(\overline{\mathrm{ABCD}}=\overline{\mathrm{EFGH}}\)

(ii) The numbers \(\overline{\mathrm{ABCDEFGH}}\) and \(\overline{\mathrm{ABCD0EFGH}}\) are both divisible by \(11\).

Let the sum of all possible values of \(\overline{\mathrm{ABCDEFGH}}\) be \(N\).

Find the digit sum of \(N\).

**Details and assumptions**:-

- \(\overline{\mathrm{ABC}}\) means the number in decimal representation with digits \(A,B,C\) i.e. \(\overline{\mathrm{ABC}} = \mathrm{100A+10B+C}\)
- The letters \(A\) to \(H\) do
**not**necessarily stand for**distinct**digits. - In the second number, the digit \(0\) is added in the middle of the 8-digit number, which makes it a 9-digit number.
- Digit sum is sum of all digits in decimal representation, digit sum of \(12023\) is \(1+2+0+2+3=8\)
- 00123 is not a 5-digit number.

Consider the sequence \( 50 + n^2 \) for positive integer \(n\):

\[51, 54, 59, 66, 75, \ldots\]

If we take the greatest common divisor of 2 consecutive terms, we obtain

\[3, 1, 1, 3, \ldots\]

What is the sum of all distinct elements in the second series?

\[ \large \frac1a+\frac1b+\frac1c= \frac1{42} \]

Let \(a\leq b \leq c\) be positive integers that satisfy the equation above. Find the maximum possible value of \(c\).

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