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Chapter 1: Q1E (page 48)


Show that in any base b鈮�2, the sum of any three single-digit numbers is at most two digits long.

Short Answer

Expert verified

Sum of any three single digit number is not as long as of a two-digit number.

Step by step solution

01

Find equation of sum of digits

Let鈥檚 assume p,q,r be the single digit numbers in base b.

Let sum of these number besumgiven as:

sum=p+q+r

Given that:

Value of p,q,rlies in range b鈮�2

Here, p,q,ris at most b鈭�1. For example: in base 10 maximum single digit is 9.

02

Calculate the value of sum

sum=p+q+r=(b鈭�1)+(b鈭�1)+(b鈭�1)=3(b鈭�1)

For b=2,

sum=3(2鈭�1)=3脳1=3

Here, sum is a single digit number.

For b=3,

sum=3(3鈭�1)=3脳2=6

Here, sum is a single digit number.

For b=9,

sum=3(9鈭�1)=3脳8=24

Here, sum is a two-digit number.

Thus, the sum is less than b2 and b2 in any base, sum is less than b2 that is represented in two digits.

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