Divide by 1-10

Number Theory Level 3

$\sum _{ x=1 }^{ 10 }{ \frac { y }{ x } } =I$

If ${ y }_{ 1 }$ is the smallest positive integer $y$ value that satisfies the equation above, ${ y }_{ 2 }$ is the second smallest, etc., then what is the value of ${ y }_{ 1 }+{ y }_{ 2 }+{ y }_{ 3 }...+{ y }_{ 100 }$ if $I$ is a positive integer?

The answer is 12726000.

1 solution

Stephen Mellor
Feb 18, 2018

$\text{Lowest common multiple} \{1,2,3,4,5,6,7,8,9,10\} = 2^3 \cdot 3^2 \cdot 5 \cdot 7 = 2520$ $\implies y_1 = 2520 \cdot 1$ $y_2 = 2520 \cdot 2$ $\vdots$ $y_n = 2520 \cdot n$ $y_100 = 2520 \cdot 100$

$y_1 + y_2 + \ldots + y_{100} = 2520 \cdot 1 + 2520 \cdot 2 + \ldots + 2520 \cdot 100$ $y_1 + y_2 + \ldots + y_{100} = 2520 \cdot (1 + 2 + 3 + \ldots + 100)$ $y_1 + y_2 + \ldots + y_{100} = 2520 \cdot \frac{100 \cdot 101}{2}$ $y_1 + y_2 + \ldots + y_{100} = 2520 \cdot 5050$ $y_1 + y_2 + \ldots + y_{100} = 12,726,000$

There's a slight problem here - it's not quite correct to look at $LCM(1,\ldots ,10)$ . It happens to work in this case, but it doesn't, for example, with the sum going up to $6$ .

The issue is the reduced form of the $n^{th}$ harmonic number. We have $H_{10}=\frac11+\frac12+\cdots +\frac{1}{10}=\frac{7381}{2520}$ . However, $H_6=\frac{49}{20}$ (in this case the fraction cancels down and the denominator is not the same as the LCM).

Chris Lewis - 1 year, 8 months ago

Divide by 1-10

The answer is 12726000.

1 solution

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