Do the following conversions from decimal number to other number systems. (i) 54 base 10 equals what in base 2? (ii) 120 base 10 equals what in base 2? (iii) 76 base 10 equals what in base 8? (iv) 889 base 10 equals what in base 8? (v) 789 base 10 equals what in base 16? (vi) 108 base 10 equals what in base 16?
Question
Do the following conversions from decimal number to other number systems.
(i) $(54)_{10}=(\text { ? })_{2} \quad$
(ii) $(120)_{10}=(\text { ? })_{2} \quad$
(iii) $(76)_{10}=(\text { ?) })_{8} \quad$
(iv) $(889)_{10}=(\text { ?) })_{8} \quad$
(v) $(789)_{10}=(\text { ? })_{16}$
(vi) $(108)_{10}=\left(\right.$ (?) ${ }_{16}$
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Answer
Let's perform the conversions step by step:
(i) $(54)_{10}$ to Binary $(?)_{2}$
To convert $54_{10}$ to binary:
Divide 54 by 2, note the remainder.
Continue dividing the quotient by 2 until the quotient is 0.
Collect the remainders from bottom to top to get the binary equivalent.
54 ÷ 2 = 27, remainder 0
27 ÷ 2 = 13, remainder 1
13 ÷ 2 = 6, remainder 1
6 ÷ 2 = 3, remainder 0
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Therefore, (54)_{10} = (110110)_{2}
(ii) $(120)_{10}$ to Binary $(?)_{2}$
To convert $120_{10}$ to binary:
Divide 120 by 2, note the remainder.
Continue dividing the quotient by 2 until the quotient is 0.
Collect the remainders from bottom to top to get the binary equivalent.
120 ÷ 2 = 60, remainder 0
60 ÷ 2 = 30, remainder 0
30 ÷ 2 = 15, remainder 0
15 ÷ 2 = 7, remainder 1
7 ÷ 2 = 3, remainder 1
3 ÷ 2 = 1, remainder 1
1 ÷ 2 = 0, remainder 1
Therefore, (120)_{10} = (1111000)_{2}
(iii) $(76)_{10}$ to Octal $(?)_{8}$
To convert $76_{10}$ to octal:
Divide 76 by 8, note the remainder.
Continue dividing the quotient by 8 until the quotient is 0.
Collect the remainders from bottom to top to get the octal equivalent.
76 ÷ 8 = 9, remainder 4
9 ÷ 8 = 1, remainder 1
1 ÷ 8 = 0, remainder 1
Therefore, (76)_{10} = (114)_{8}
(iv) $(889)_{10}$ to Octal $(?)_{8}$
To convert $889_{10}$ to octal:
Divide 889 by 8, note the remainder.
Continue dividing the quotient by 8 until the quotient is 0.
Collect the remainders from bottom to top to get the octal equivalent.
889 ÷ 8 = 111, remainder 1
111 ÷ 8 = 13, remainder 7
13 ÷ 8 = 1, remainder 5
1 ÷ 8 = 0, remainder 1
Therefore, (889)_{10} = (1571)_{8}
(v) $(789)_{10}$ to Hexadecimal $(?)_{16}$
To convert $789_{10}$ to hexadecimal:
Divide 789 by 16, note the remainder.
Continue dividing the quotient by 16 until the quotient is 0.
Collect the remainders from bottom to top to get the hexadecimal equivalent.
789 ÷ 16 = 49, remainder 13 (D)
49 ÷ 16 = 3, remainder 1
3 ÷ 16 = 0, remainder 3
Therefore, (789)_{10} = (31D)_{16}
(vi) $(108)_{10}$ to Hexadecimal $(?)_{16}$
To convert $108_{10}$ to hexadecimal:
Divide 108 by 16, note the remainder.
Continue dividing the quotient by 16 until the quotient is 0.
Collect the remainders from bottom to top to get the hexadecimal equivalent.
108 ÷ 16 = 6, remainder 12 (C)
6 ÷ 16 = 0, remainder 6
Therefore, (108)_{10} = (6C)_{16}
Summary
$(54)_{10} = (110110)_{2}$
$(120)_{10} = (1111000)_{2}$
$(76)_{10} = (114)_{8}$
$(889)_{10} = (1571)_{8}$
$(789)_{10} = (31D)_{16}$
$(108)_{10} = (6C)_{16}$
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