Multiplication in base five

Multiplication in base five can be challenging. However, I will try my best to break the information down into into steps to make it easy to understand and master.

It is similar to multiplying numbers in base 10. However, because we are not used to do this in base 5, it may appear tough.

Plan to spend some time reading this lesson about multiplication in base five more than once.

First, know the difference between base 5 place value and base 10 place value.

I show the difference with two numbers: 48573 and 134125

For 48573, 5 is in the hundreds place and it means there are 5 hundreds

For 134125, 4 is in the twenty-fives place and it means there are 4 twenty-fives

Difference between base 10 and base 5

The most difficult concept with multiplication in base five is to master how to carry numbers.

Study the following two multiplications carefully. The one on the right is the multiplication in base five.

Example showing how to do multiplication in base five

Multiplication in base five

         8 6 4
      ×    3 6
   _____________
         3 4 35
      ×    2 45
   ______________


  5 3 2
       8 6 4
    ×    3 6
  __________
   5 1 8 4
   3 3 2
       3 4 35
    ×    2 45
  ___________
   3 0 3 25
Step 1:

6 × 4 = 24.

24 = 10 + 10 + 4

You have 2 tens to carry.

Write 4 down in the ones place.

Carry 2 tens by putting 2 in the tens place (Shown in red)
Step 1:

4 × 3 = 12.

12 = 5 + 5 + 2

You have 2 fives to carry.

Write 2 down in the ones place.

Carry 2 fives by putting 2 in the fives place (Shown in red)


Step 2:

6 × 6 = 36.

36 + 2 = 38 tens

38 tens = 10 tens + 10 tens + 10 tens + 8 tens

= 100 + 100 + 100 + 80

= 100 + 100 + 100 + 8 tens

You have 3 hundreds to carry. Write 8 down in the tens place and carry 3 hundreds by putting 3 in the hundreds place (Shown in green)
Step 2:

4 × 4 = 16.

16 + 2 = 18 fives

18 fives = 5 fives + 5 fives + 5 fives + 3 fives

= 25 + 25 + 25 + 15

= 25 + 25 + 25 + 3 fives

You have 3 twenty-fives to carry. Write 3 down in the fives place and carry 3 twenty-fives by putting 3 in the twenty-fives place (Shown in green)


Step 3:

6 × 8 = 48.

48 + 3 = 51 hundreds

51 hundreds = 10 hundreds + 10 hundreds + 10 hundreds + 10 hundreds + 10 hundreds + 1 hundred

= 1000 + 1000 + 1000 + 1000 + 1000 + 100

= 1000 + 1000 + 1000 + 1000 + 1000 + 1 hundred

You have 5 thousands to carry. Write 1 down in the hundreds place and carry 5 thousands by putting 5 in the thousands place (Shown in black)

Bring down the 5 in the thousands place
Step 3:

4 × 3 = 12.

12 + 3 = 15 twenty-fives

15 twenty-fives = 5 twenty-fives + 5 twenty-fives + 5 twenty-fives + 0 twenty-five

= 125 + 125 + 125 + 0

= 125 + 125 + 125 + 0 twenty-five

You have 3 one hundred twenty-fives to carry. Write 0 down in the twenty fives place and carry 3 one hundred twenty-fives by putting 3 in the one hundred twenty-five place (Shown in black)

Bring down the 3 in the one hundred twenty-fives place.

Notice again the 0 in the twenty-fives place.




   
   2 1 1
           8 6 4
           ×  3 0
  ________________
   2 5 9 2 0
   1 1 1
            3 4 35
            ×  2 05
   _______________
    1 2 4 1 05
Step 4:

3 × 4 = 12 tens.
12 tens = 10 tens + 2 tens = 100 + 2 tens

Carry 1 hundred by putting 1 in the hundreds place (Shown in red)

Write 2 down in the tens place.

Since there is nothing in the ones place, just put a 0.
Step 4:

2 × 3 = 6 fives.
6 fives = 5 fives + 1 five = 25 + 1 five

Carry 1 twenty five by putting 1 in the twenty-fives place (Shown in red)

Write 1 down in the fives place.

Since there is nothing in the ones place, just put a 0.
Step 5:

3 × 6 = 18 hundreds.

18 hundreds + 1 hundred = 19 hundreds

19 hundreds = 10 hundreds + 9 hundreds

= 1000 + 9 hundreds

Write 9 down in the hundreds place and carry 1 thousand by putting 1 in the thousand place (Shown in green)

Step 5:

2 × 4 = 8 twenty-fives.

8 twenty-fives + 1 twenty-five = 9 twenty-fives

9 twenty-fives = 5 twenty-fives + 4 twenty-fives

= 125 + 4 twenty-fives

Write 4 down in the twenty-fives place and carry 1 one hundred twenty-five by putting 1 in the one hundred twenty-five place (Shown in green)

Step 6:

3 × 8 = 24 thousands.

24 thousands + 1 thousand = 25 thousands

25 thousands = 10 thousands + 10 thousands + 5 thousands

= 10000 + 10000 + 5 thousands

Write 5 down in the thousands place and carry 2 ten thousands by putting 2 in the ten thousands place (Shown in black)

Bring the 2 down in the ten thousands place
Step 6:

2 × 3 = 6 one hundred twenty-fives.

6 one hundred twenty-fives + 1 one hundred twenty five = 7 one hundred twenty-fives

7 one hundred twenty-fives = 5 one hundred twenty-fives + 2 one hundred twenty-five = 625 + 2 one hundred twenty-five

Write 2 down in the one hundred twenty-fives place and carry 1 six hundred twenty-five by putting 1 in the six hundred twenty-five place (Shown in black)



Just add now. You may need to review addition in base five before even starting this lesson about multiplication in base five.

Adding the results in base ten.

  1 1 1
       5 1 8 4
   2 5 9 2 0
  ____________
   3 1 1 0 4


Adding the results in base five.

  1
       3 0 3 2
   1 2 4 1 0
  ______________
   2 0 4 4 2

Again, it is quite challenging to do multiplication in base five. I recommend you take your time when reading this lesson. Don't give up!

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