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 5                 5 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

Browser mismatches may mean the lines dont’t align vertically. I do not yet see how to fix this, short of turning the calculation into an image.

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.