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When the base-b number 11011_b is multiplied by b-1, then 1001_b is added, what is the result (written in base b)?

Guest Aug 17, 2018
#1
+1

The number you have listed, 11011_b can be represented in ANY base from 2 and up. When you say "is multiplied by b-1", I'm assuming you mean: (11011_b) - 1. If I'm wrong in my assumptions, let us know.

11011*11010+1001 =1011000111.00 - this is in base 2
=122002111.00 - this is in base 3
=121232111.00 - this is in base 4, 5, 6, 7, 8, 9,.........etc.

Note: You may choose the base you want, or is intended.

Guest Aug 17, 2018
#2
+1

I don't think the previous answerer got the question right. Let's take a particular example, say b=4. Then the question becomes "When the base 4 number $$11011_4$$ is multiplied by 3, and then $$1001_4$$ is added, what is the result (written in base 4)?"

Now$$11011_4\times 3 = 33033_4$$ and adding $$1001_4$$ we get $$100100_4$$.

Similarly, in for example base 7, we have

$$11011_7\times 6 + 1001_7 = 66066_7+1001_7=100100_7$$

In fact for any base>1 the answer is always 100100.

Guest Aug 22, 2018
#3
+93866
+2

Yes the second answer is definitely the correct one.

I am glad you gave it a go guest one, that is the best way to learn.  I hope you see this continuation.

Good work guest 2

I shall also try to explain.

When the base-b number 11011_b is multiplied by b-1, then 1001_b is added, what is the result (written in base b)?

$$11011_b=b^4+b^3+0+b+1\\ (b-1)*(b^4+b^3+0+b+1)\\ =b^5+b^4+0+b^2+b\\ \quad\;\;-(b^4+b^3+0+b+1)\\ =b^5+0-b^3+b^2+0-1\\ Now\\ 1001_b=b^3+0+0+1\\~\\ \;\;\;(b^5+0-b^3+b^2+0-1)\\ \;\;\;+\qquad\;\;(b^3+0+0+1)\\ =b^5+0+0+b^2+0+0\\ =100100_b$$

Melody  Aug 22, 2018
edited by Melody  Aug 22, 2018
#4
+93866
+2

If you are having trouble understanding what I have done I shall try to compare it to base 10

Remember:

In base 10

$$23606 = 2*10^4+3*10^3+6*10^2+0*10+6$$

or

$$110100\\ = 1*10^5+1*10^4+0+1*10^2+0+0 \\ = 10^5+10^4+0+10^2+0+0$$

Melody  Aug 22, 2018