Jelly, 11 bytes

LÆḌs@^/ẸɗÐḟ

Try it online!

Input as a list of bits, the Footer does this for you

How it works

LÆḌs@^/ẸɗÐḟ - Main link. Takes a list of bits, B, on the left
L - Length of B
 ÆḌ - Proper divisors
 ɗÐḟ - Filter proper divisors k, keeping those which return False:
 s@ - Split B into pieces of length k
 ^/ - Reduce columnwise by XOR
 Ẹ - Are any true?

Jelly, 11 bytes

LÆḌs@^/ẸɗÐḟ

Try it online!

Input as a list of bits, the Footer does this for you

How it works

LÆḌs@^/ẸɗÐḟ - Main link. Takes a list of bits, B, on the left
L - Length of B
 ÆḌ - Proper divisors
 ɗÐḟ - Filter proper divisors k, keeping those which return False:
 s@ - Split B into pieces of length k
 ^/ - Reduce columnwise by XOR
 Ẹ - Are any true?

Monads you could use instead of Ẹ:

• S: Sum of columns
• T: Indexes of non-zero elements
• Ḅ: Convert from binary
• Ḍ: Convert from decimal
• Ṭ: Generate an array which would yield the original argument under T
• Ṁ: Maximum
• §: Sum of rows

Jelly, 11 bytes

LÆḌs@^/ẸɗÐḟ

Try it online!

Input as a list of bits, the Footer does this for you

Jelly, 11 bytes

LÆḌs@^/ẸɗÐḟ

Try it online!

Input as a list of bits, the Footer does this for you

How it works

LÆḌs@^/ẸɗÐḟ - Main link. Takes a list of bits, B, on the left
L - Length of B
 ÆḌ - Proper divisors
 ɗÐḟ - Filter proper divisors k, keeping those which return False:
 s@ - Split B into pieces of length k
 ^/ - Reduce columnwise by XOR
 Ẹ - Are any true?