Question

identical language in explanations will not be accepted. Math 214: Project 2 1. Multiplication patterns in other 5 bases In base 6, when we add 5+ 5 14, -3..2. 52 5, 3 5 we get 14 If we add on another 5 we get 23 Add on another 5, to find 5, 4 b) Complete the base 6 multiplication table below, using your result, above, and the patterns you see: e) Color or shade the column that shows the multiples of 5. Describe at least two things you notice about the pattern that the multiples of 5 make in base 6 multiplication. I.All the multiples of 5 have a 5 or a zero in the ones places 2. The multiples 10 and 20 are both across from each other 4) Complete the base 8 multiplication table below, using the patterns you see and the fact that multiplication is commutative (4 5-5 4) Color or shade the column that shows the multiples of 7. How is this pattern similar to the multiples of 5 in base 6? I.This pattern is similar because the number itself (7) and (5)

Answer 1

When doing addition in base 6, increase the number of digits as soon as 6 is reached.

For example, while adding 4₆ and 1₆, 4₆ + 1₆ = 5_6, but adding one more 1₆, we get 5₆ + 1₆ = 10₆, because a 6 is reached and we have to increase the number of digits.

Similarly, 5₆ + 5₆ can be written as (5₆ + 1₆) + 4₆ = 10₆ + 4₆ = 14₆. And similar process is carried on.

Now in a.)

5₆* 4₆ = 23₆ + 4₆ = (23₆ + 3₆) + 2₆ = 30₆ + 2₆ = 32₆

b)

0	1	2	3	4	5
1	1	2	3	4	5
2	2	4	10	12	14
3	3	10	13	20	23
4	4	12	20	24	32
5	5	14	23	32	41

• 5*5 = 25 = 24+1 = 4*6¹ + 1 = 41₆
• 5*4 = 20 = 18 + 2 = 3*6 + 2 = 31₆
• 5*3 = 15 = 12 + 3 = 2*6 + 3 = 23₆
• 5*2 = 10 = 1*6 + 4 = 14₆
• 5*1 = 5 = 5₆
• 4*4 = 16 = 2*6 + 4 = 24₆
• 4*3 = 12 = 2*6 = 20₆
• 4*2 = 8 = 6+2 = 12₆
• 4*1 = 4₆
• 3*3 = 9 = 6+3 = 13₆
• 3*2 = 6 = 10₆
• 3*1 = 3₆
• 2*2 = 4 = 4₆

c.) Multiples of 5 have been bolded in the table.

• The sum of the figits in the number is equal to 5, or multiple of 5.
• Digits are reversed after 3*2.
• They lie on the edge of the table.

d.)

0	1	2	3	4	5	6	7
1	1	2	3	4	5	6	7
2	2	4	6	10	12	14	16
3	3	6	11	14	17	22	25
4	4	10	14	20	24	30	34
5	5	12	17	24	31	36	43
6	6	14	22	30	36	44	52
7	7	16	25	34	43	52	61

e.) Multiple of 7 have been bolded. We notice that -

• The sum of the figits in the number is equal to 7, or multiple of 7.
• Digits are reversed after 4*7.
• They lie on the edge of the table.