#Pyth, 21 20 18 bytes

hoacFN.n3C_Bf!%QTS 

Try it online. Test suite.

###Explanation

1. Get the incluSive range from 1 to input.
2. filter for numbers for that divide the input !%QT.
3. Get [that list, that list reversed] _B. The reversed divisors of a number is the same list as the number divided by each of the divisors.
4. Transpose the list to get pairs of [numerator, denominator].
5. Sort the pairs by the absolute difference of the ratio of the pair cFN and the golden ratio .n3.
6. Get the first (lowest) pair h and print.

Pyth, 21 20 18 bytes

hoacFN.n3C_Bf!%QTS 

Try it online. Test suite.

Explanation

1. Get the incluSive range from 1 to input.
2. filter for numbers for that divide the input !%QT.
3. Get [that list, that list reversed] _B. The reversed divisors of a number is the same list as the number divided by each of the divisors.
4. Transpose the list to get pairs of [numerator, denominator].
5. Sort the pairs by the absolute difference of the ratio of the pair cFN and the golden ratio .n3.
6. Get the first (lowest) pair h and print.

#Pyth, 21 20 18 bytes

hoacFN.n3C_Bf!%QTS 

Try it online. Test suite.

#Pyth, 21 20 18 bytes

hoacFN.n3C_Bf!%QTS 

Try it online. Test suite.

###Explanation

1. Get the incluSive range from 1 to input.
2. filter for numbers for that divide the input !%QT.
3. Get [that list, that list reversed] _B. The reversed divisors of a number is the same list as the number divided by each of the divisors.
4. Transpose the list to get pairs of [numerator, denominator].
5. Sort the pairs by the absolute difference of the ratio of the pair cFN and the golden ratio .n3.
6. Get the first (lowest) pair h and print.

#Pyth, 21 20 bytes

hoacFN.n3,R/Qdf!%QTS 

Try it online. Test suite.

#Pyth, 21 20 18 bytes

hoacFN.n3C_Bf!%QTS 

Try it online. Test suite.