Solve 1/56+1/57+1/58+1/59+frac{1}{59}+frac{1}{60}+frac{1}{61}+frac{1}{61}+frac{1}{61}+frac{1}{61}+frac{1}{61}+frac{1}{62}+frac{1}{64}+frac{1}{65}+frac{1}{66}+frac{1}{68}

$$\frac{ 1 }{ 56 } + \frac{ 1 }{ 57 } + \frac{ 1 }{ 58 } + \frac{ 1 }{ 59 } + \frac{ 1 }{ 59 } + \frac{ 1 }{ 60 } + \frac{ 1 }{ 61 } + \frac{ 1 }{ 61 } + \frac{ 1 }{ 61 } + \frac{ 1 }{ 61 } + \frac{ 1 }{ 61 } + \frac{ 1 }{ 62 } + \frac{ 1 }{ 64 } + \frac{ 1 }{ 65 } + \frac{ 1 }{ 66 } + \frac{ 1 }{ 68 }$$

Evaluate

$$\frac{52657880485909}{200853482846016}\approx 0.262170612$$

Solution Steps

Least common multiple of 56 and 57 is 3192. Convert \frac{1}{56} and \frac{1}{57} to fractions with denominator 3192.

$$\frac{57}{3192}+\frac{56}{3192}+\frac{1}{58}+\frac{1}{59}+\frac{1}{59}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{57}{3192} and \frac{56}{3192} have the same denominator, add them by adding their numerators.

$$\frac{57+56}{3192}+\frac{1}{58}+\frac{1}{59}+\frac{1}{59}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 57 and 56 to get 113.

$$\frac{113}{3192}+\frac{1}{58}+\frac{1}{59}+\frac{1}{59}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 3192 and 58 is 92568. Convert \frac{113}{3192} and \frac{1}{58} to fractions with denominator 92568.

$$\frac{3277}{92568}+\frac{1596}{92568}+\frac{1}{59}+\frac{1}{59}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{3277}{92568} and \frac{1596}{92568} have the same denominator, add them by adding their numerators.

$$\frac{3277+1596}{92568}+\frac{1}{59}+\frac{1}{59}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 3277 and 1596 to get 4873.

$$\frac{4873}{92568}+\frac{1}{59}+\frac{1}{59}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 92568 and 59 is 5461512. Convert \frac{4873}{92568} and \frac{1}{59} to fractions with denominator 5461512.

$$\frac{287507}{5461512}+\frac{92568}{5461512}+\frac{1}{59}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{287507}{5461512} and \frac{92568}{5461512} have the same denominator, add them by adding their numerators.

$$\frac{287507+92568}{5461512}+\frac{1}{59}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 287507 and 92568 to get 380075.

$$\frac{380075}{5461512}+\frac{1}{59}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 5461512 and 59 is 5461512. Convert \frac{380075}{5461512} and \frac{1}{59} to fractions with denominator 5461512.

$$\frac{380075}{5461512}+\frac{92568}{5461512}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{380075}{5461512} and \frac{92568}{5461512} have the same denominator, add them by adding their numerators.

$$\frac{380075+92568}{5461512}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 380075 and 92568 to get 472643.

$$\frac{472643}{5461512}+\frac{1}{60}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 5461512 and 60 is 27307560. Convert \frac{472643}{5461512} and \frac{1}{60} to fractions with denominator 27307560.

$$\frac{2363215}{27307560}+\frac{455126}{27307560}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{2363215}{27307560} and \frac{455126}{27307560} have the same denominator, add them by adding their numerators.

$$\frac{2363215+455126}{27307560}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 2363215 and 455126 to get 2818341.

$$\frac{2818341}{27307560}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Reduce the fraction \frac{2818341}{27307560} to lowest terms by extracting and canceling out 3.

$$\frac{939447}{9102520}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 9102520 and 61 is 555253720. Convert \frac{939447}{9102520} and \frac{1}{61} to fractions with denominator 555253720.

$$\frac{57306267}{555253720}+\frac{9102520}{555253720}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{57306267}{555253720} and \frac{9102520}{555253720} have the same denominator, add them by adding their numerators.

$$\frac{57306267+9102520}{555253720}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 57306267 and 9102520 to get 66408787.

$$\frac{66408787}{555253720}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 555253720 and 61 is 555253720. Convert \frac{66408787}{555253720} and \frac{1}{61} to fractions with denominator 555253720.

$$\frac{66408787}{555253720}+\frac{9102520}{555253720}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{66408787}{555253720} and \frac{9102520}{555253720} have the same denominator, add them by adding their numerators.

$$\frac{66408787+9102520}{555253720}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 66408787 and 9102520 to get 75511307.

$$\frac{75511307}{555253720}+\frac{1}{61}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 555253720 and 61 is 555253720. Convert \frac{75511307}{555253720} and \frac{1}{61} to fractions with denominator 555253720.

$$\frac{75511307}{555253720}+\frac{9102520}{555253720}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{75511307}{555253720} and \frac{9102520}{555253720} have the same denominator, add them by adding their numerators.

$$\frac{75511307+9102520}{555253720}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 75511307 and 9102520 to get 84613827.

$$\frac{84613827}{555253720}+\frac{1}{61}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 555253720 and 61 is 555253720. Convert \frac{84613827}{555253720} and \frac{1}{61} to fractions with denominator 555253720.

$$\frac{84613827}{555253720}+\frac{9102520}{555253720}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{84613827}{555253720} and \frac{9102520}{555253720} have the same denominator, add them by adding their numerators.

$$\frac{84613827+9102520}{555253720}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 84613827 and 9102520 to get 93716347.

$$\frac{93716347}{555253720}+\frac{1}{61}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 555253720 and 61 is 555253720. Convert \frac{93716347}{555253720} and \frac{1}{61} to fractions with denominator 555253720.

$$\frac{93716347}{555253720}+\frac{9102520}{555253720}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{93716347}{555253720} and \frac{9102520}{555253720} have the same denominator, add them by adding their numerators.

$$\frac{93716347+9102520}{555253720}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 93716347 and 9102520 to get 102818867.

$$\frac{102818867}{555253720}+\frac{1}{62}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 555253720 and 62 is 17212865320. Convert \frac{102818867}{555253720} and \frac{1}{62} to fractions with denominator 17212865320.

$$\frac{3187384877}{17212865320}+\frac{277626860}{17212865320}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{3187384877}{17212865320} and \frac{277626860}{17212865320} have the same denominator, add them by adding their numerators.

$$\frac{3187384877+277626860}{17212865320}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 3187384877 and 277626860 to get 3465011737.

$$\frac{3465011737}{17212865320}+\frac{1}{64}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 17212865320 and 64 is 137702922560. Convert \frac{3465011737}{17212865320} and \frac{1}{64} to fractions with denominator 137702922560.

$$\frac{27720093896}{137702922560}+\frac{2151608165}{137702922560}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{27720093896}{137702922560} and \frac{2151608165}{137702922560} have the same denominator, add them by adding their numerators.

$$\frac{27720093896+2151608165}{137702922560}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Add 27720093896 and 2151608165 to get 29871702061.

$$\frac{29871702061}{137702922560}+\frac{1}{65}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 137702922560 and 65 is 1790137993280. Convert \frac{29871702061}{137702922560} and \frac{1}{65} to fractions with denominator 1790137993280.

$$\frac{388332126793}{1790137993280}+\frac{27540584512}{1790137993280}+\frac{1}{66}+\frac{1}{68}$$

Since \frac{388332126793}{1790137993280} and \frac{27540584512}{1790137993280} have the same denominator, add them by adding their numerators.

$$\frac{388332126793+27540584512}{1790137993280}+\frac{1}{66}+\frac{1}{68}$$

Add 388332126793 and 27540584512 to get 415872711305.

$$\frac{415872711305}{1790137993280}+\frac{1}{66}+\frac{1}{68}$$

Reduce the fraction \frac{415872711305}{1790137993280} to lowest terms by extracting and canceling out 5.

$$\frac{83174542261}{358027598656}+\frac{1}{66}+\frac{1}{68}$$

Least common multiple of 358027598656 and 66 is 11814910755648. Convert \frac{83174542261}{358027598656} and \frac{1}{66} to fractions with denominator 11814910755648.

$$\frac{2744759894613}{11814910755648}+\frac{179013799328}{11814910755648}+\frac{1}{68}$$

Since \frac{2744759894613}{11814910755648} and \frac{179013799328}{11814910755648} have the same denominator, add them by adding their numerators.

$$\frac{2744759894613+179013799328}{11814910755648}+\frac{1}{68}$$

Add 2744759894613 and 179013799328 to get 2923773693941.

$$\frac{2923773693941}{11814910755648}+\frac{1}{68}$$

Least common multiple of 11814910755648 and 68 is 200853482846016. Convert \frac{2923773693941}{11814910755648} and \frac{1}{68} to fractions with denominator 200853482846016.

$$\frac{49704152796997}{200853482846016}+\frac{2953727688912}{200853482846016}$$

Since \frac{49704152796997}{200853482846016} and \frac{2953727688912}{200853482846016} have the same denominator, add them by adding their numerators.

$$\frac{49704152796997+2953727688912}{200853482846016}$$

Add 49704152796997 and 2953727688912 to get 52657880485909.

$$\frac{52657880485909}{200853482846016}$$

Factor

$$\frac{3257 \cdot 36479 \cdot 443203}{2 ^ {6} \cdot 3 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 29 \cdot 31 \cdot 59 \cdot 61} = 0.26217061182991325$$