Number Theory

# Number Theory Warmups: Level 5 Challenges

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

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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