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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. See more

Level 5

         

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

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?

If we write out the decimal expansion of the smaller root of \(f(x)=1000000x^2-1000000x+1\), we get the following decimal: \[0.000001 \quad 000001\quad 000002\quad 000005 \quad 000014 \quad \ldots\] which we notice as the first 5 Catalan numbers. How many distinct Catalan numbers occur in a row before this pattern stops?

Image Credit: Wikimedia Dmharvey.

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

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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