vera standifer

? how do you change a base 10 number to base 5

this is 6th grade middle school level....I am a perplexed paren who have been out of school for 20 years....please help



Hi Vera,

When we use base 10 to express a number, say 5472, we are doing so in powers of 10. That is

5472 = 5x1000 + 4x100 + 7x10 +2
= 5x103 + 4x102 + 7x10 + 2

To express a number in base 5 notation you need to write it in powers of 5. Let me take 613 as an example. First I want to see how many digits I will need. To do this I calculate powers of 5

51 = 5

52 = 25

53 = 125

54 = 625 625 is larger than 613 so I only need a 53 digit, a 52 digit, a 5 digit and a units digit. Thus I will have a 4 digit, base 5 number.

Start with the 53 digit. How many 125's are there in 613?

 613/125 = 4 with a remainder of 113 Thus 613 = 4x125 + 113 = 4x53 + 113 Hence the first digit (reading left to right) is a 4.

The remainder is 113 so how many 25's are there in 113?

 113/25 = 4 with a remainder of 13 Thus 613 = 4x125 + 4x25 + 13 = 4x53 + 4x52 + 13 Hence the second digit (reading left to right) is also a 4.

The remainder is 13 so how many 5's are there in 13?

 13/5 = 2 with a remainder of 3 Thus 613 = 4x125 + 4x25 + 2x5 + 3 = 4x53 + 4x52 + 2x5 + 3 Hence the third digit (reading left to right) is a 2 and the units digit is 3. Therefore 613 written in base 5 is 4423 I hope this helps,
Penny



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