Answer to Question #311841 in Discrete Mathematics for sji

Question

  Find the product of 10101₂ and 101₂.  

Answer 1

Solution

One of the easiest ways to find the product of the two binary numbers is to first convert them to decimal, multiply them and convert the final answer back to binary number.

Now,

$(10101)_2=(1)\times 2^4+(0)\times 2^3+(1)\times 2^2+(0)\times 2^1+(1)\times 2^0$

$(10101)_2=(1)\times (16)+(0)\times (8)+(1)\times (4)+(0)\times (2)+(1)\times (1)$

$(10101)_2=16+0+4+0+1$

$(10101)_2=(21)$₁₀

Now for the second number

$(101)_2=(1)\times 2^2+(0)\times 2^1+(1)\times 2^0$

$(101)_2=(1)\times (4)+(0)\times (2)+(1)\times (1)$

$(101)_2=4+0+1$

$(101)_2=(5)_{10}$

Now,

$(10101)_2\times(101)_2=21\times5$

$(10101)_2\times(101)_2=(105)_{10}$

Now we convert back $(105)$₁₀ to binary, which is

$(105)_{10} = (1101001)_2$

Hence

$(10101)_2\times(101)_2= (01101001)_2$