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# Greatest Common Divisor / Lowest Common Multiple

What is the largest number that can divide two numbers without a remainder? What is the smallest number that is divisible by two numbers without a remainder?

# Greatest Common Divisor / Lowest Common Multiple: Level 4 Challenges

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?

Find the sum of all integers $$k$$ with $$1\leq k\leq 2015$$ and $$\gcd(k,2015)=1$$.

Let $$m$$ and $$n$$ be positive integers such that 11 divides $$m+13n$$ and 13 divides $$m+11n$$.

What is the minimum value of $$m+n$$?

$\text{lcm}(1,77) + \text{lcm}(2,77) + \text{lcm}(3,77) + \cdots + \text{lcm}(77,77) = \, ?$

Notation: $$\text{lcm}(a,b)$$ denote the Lowest Common Multiple of $$a$$ and $$b$$.

$\large\underbrace{0,1,0,1,0,0,0,1,0,1}_{\text{period}}, 0,1,0,1,0,0,0,1,0,1,0,\dots$ Let $$n$$ be a positive integer, and let the function $$f_n:\ \mathbb N \to \{0,1\}$$ be defined by $f_n(m) = \begin{cases} 0 & \gcd(n,m) > 1 \\ 1 & \gcd(n,m) = 1. \end{cases}$ That is, $$f_n(m)$$ tells us whether $$m$$ is coprime to $$n$$ or not. If we write out the values of $$f_n,$$ we get a periodic (repeating) pattern. For instance, the list above gives the values of $$f_{10}$$ starting at $$m = 0$$; the pattern repeats itself with period 10.

Consider the function $$f_{1400}$$. What is its period?


Note: The period is defined as the smallest possible value, that is, the least positive $$T$$ such that $$f_n(m + T) = f_n(m)$$ for all integers $$m$$.

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