Answer:
[tex](a)\ 222_{10} = DE_{16}[/tex] --- True
[tex](b)\ D7_{16} = 11010011_2[/tex] --- False
[tex](c)\ 13_{16} = 19_{10}[/tex] --- True
Explanation:
Required
Determine if the statements are true or not.
[tex](a)\ 222_{10} = DE_{16}[/tex]
To do this, we convert DE from base 16 to base 10 using product rule.
So, we have:
[tex]DE_{16} = D * 16^1 + E * 16^0[/tex]
In hexadecimal.
[tex]D =13 \\E = 14[/tex]
So, we have:
[tex]DE_{16} = 13 * 16^1 + 14 * 16^0[/tex]
[tex]DE_{16} = 222_{10}[/tex]
Hence:
(a) is true
[tex](b)\ D7_{16} = 11010011_2[/tex]
First, convert D7 to base 10 using product rule
[tex]D7_{16} = D * 16^1 + 7 * 16^0[/tex]
[tex]D = 13[/tex]
So, we have:
[tex]D7_{16} = 13 * 16^1 + 7 * 16^0[/tex]
[tex]D7_{16} = 215_{10}[/tex]
Next convert 215 to base 2, using division rule
[tex]215 / 2 = 107 R 1[/tex]
[tex]107/2 =53 R 1[/tex]
[tex]53/2 =26 R1[/tex]
[tex]26/2 = 13 R 0[/tex]
[tex]13/2 = 6 R 1[/tex]
[tex]6/2 = 3 R 0[/tex]
[tex]3/2 = 1 R 1[/tex]
[tex]1/2 = 0 R1[/tex]
Write the remainders from bottom to top.
[tex]D7_{16} = 11010111_2[/tex]
Hence (b) is false
[tex](c)\ 13_{16} = 19_{10}[/tex]
Convert 13 to base 10 using product rule
[tex]13_{16} = 1 * 16^1 + 3 * 16^0[/tex]
[tex]13_{16} = 19[/tex]
Hence; (c) is true
