Number Theory

Number Theory Warmups

Number Theory Warmups: Level 5 Challenges


2122+2324+2526+27+22013 2^{1}-2^{2}+2^{3}-2^{4}+2^{5}-2^{6}+2^{7} -\ldots +2^{2013}

Let aa be the value of the expression above. Find the last two digits of a.a.

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

(i) ABCD=EFGH\overline{\mathrm{ABCD}}=\overline{\mathrm{EFGH}}
(ii) The numbers ABCDEFGH\overline{\mathrm{ABCDEFGH}} and ABCD0EFGH\overline{\mathrm{ABCD0EFGH}} are both divisible by 1111.
Let the sum of all possible values of ABCDEFGH\overline{\mathrm{ABCDEFGH}} be NN.

Find the digit sum of NN.

Details and assumptions:-

  • ABC\overline{\mathrm{ABC}} means the number in decimal representation with digits A,B,CA,B,C i.e. ABC=100A+10B+C\overline{\mathrm{ABC}} = \mathrm{100A+10B+C}
  • The letters AA to HH do not necessarily stand for distinct digits.
  • In the second number, the digit 00 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 1202312023 is 1+2+0+2+3=81+2+0+2+3=8
  • 00123 is not a 5-digit number.

Consider the sequence 50+n2 50 + n^2 for positive integer nn:

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

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

3,1,1,3,3, 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)=1000000x21000000x+1f(x)=1000000x^2-1000000x+1, we get the following decimal: 0.0000010000010000020000050000140.000001 \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.

1a+1b+1c=142 \large \frac1a+\frac1b+\frac1c= \frac1{42}

Let abca\leq b \leq c be positive integers that satisfy the equation above. Find the maximum possible value of cc.


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