Multiplication by hand is quite difficult. When you learned to do this, (Year 5? In denary) you laid the calculation out in a way that multiplied the top number by each digit of the lower number—with its place preserved.
Multiply 235 by 37 as an exercise (in base ten): shown below as the first example. What you do is 7x235 and 30x235 and add the parts. How do you do 7x235 without a calculator? “7x5 is 35 so 5 down carry 3; 7x3 (really 7x30) is 21 and add the 3 carried makes 24; 4 down carry 2; 7x2 is 14 add the carried 2 is 16; answer is 1645.” Hard, isn’t it?
So how do you do 235x7 in base eight? [Third example] 7x5=3510=438 so 3 down, 4 to carry; 7x3=2110=258 add 4 is 318 so 1 down 3 to carry; 7x2=1410 plus the carried 3 is 1710=218 so 21138 is the answer.
Below, I have given you 5532x37 in denary and octal to study; You might make the working relate to the exercise....
2 3 5 5 5 3 2 2 3 55 5 3 2
x 7 10 x 3 7 10. x 7 8 x 3 7. 8
2 3 3 3 2 1 3 4 4 4 2 1
1 6 4 5 3 8 7 2 4 2 1 1 3 4 4 5 5 6
1 2 2 1
1 6 5 9 6 0 2 1 0 1 6 0
1 1 . 1
2 0 4 6 8 4 2 5 4 7 3 6
Exercise:
1 In base 8, do 235x3, 37x2, 37x3, 37x5. From this do 378x2358. You should get 114038.
2 In base 8, do 325x7, 325x4 and hence 325x74. Similarly do 74x2, 74x3, 74x5 and 74x325.
3 In base 4, do 321x3, 321x2 and hence 321x32 and 321x23.
4 Assuming your last answer is right, rewrite question and answer in base 16 (hint 16 = 42)
5 In base 6, do 123x4, 123x5 and hence 123x45 and 123x54.
6 Convert 5532 x 37 from base 8 to base 2 (378 = 011 1112) and write the answer too.
7 Assuming your last is correct, convert the problem and answer to base 4, quartenary.
8 Assuming your last is correct, convert the problem and answer to base 16, hexadecimal.
9 Multiply ABBA by F017. The base is obvious.
10 Calculate te712 x 8et12 This is duodecimal.
11Multiply 2212 by itself in bases two and five.