Suppose I have two binary numbers which has a combination of both zeroes and ones. Is it possible to show that such two binary numbers which have at least one zero in it can never be multiplied to give a binary number with no zero in it ( i.e. , the resulting binary number is a combination of only 1's ) ?
2026-03-31 04:17:28.1774930648
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Multiplication of two binary numbers
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Minimal counterexample: the binary number $11111111$.
Indeed, one has $(110011)_2 \times (101)_2 = (11111111)_2 = (51)_{10} \times (5)_{10} = (255)_{10}$ and the smaller binary numbers containing only $1$'s cannot be decomposed as a product of binary numbers containing a $0$: $(11)_2 = (3)_{10}, (111)_2 = (7)_{10}, (11111)_2 = (31)_{10}, (1111111)_2 = (127)_{10}$ are prime, $(1111)_2 = (15)_{10} = (11)_2 \times (101)_2$ and $(111111)_2 = (63)_{10} = (111)_2 \times (1001)_2$.
No. Try $23\times89=2047$ in binary: $10111_2\times1011001_2=11111111111_2$.