+0

+1
150
9

1)  If 554 base b is the base  b representation of the square of the number whose base  b representation is  24 base b then find b.

2)   A math textbook with a double-digit number of pages is split into sections. Each section is exactly 12 pages long, with the exception of the epilogue, which is  11 pages long. Every page belongs to a section. Furthermore, on the bottom of each 5th page, a trivia fact is presented on the bottom of the page, starting from the fifth page. If a trivia fact appears on the bottom of the second-to-last page, then how many pages does the textbook have?

3)  When a positive integer is expressed in base 7, it is AB base 7 , and when it is expressed in base 5, it is  BA base 5. What is the positive integer in decimal?

May 27, 2019

#1
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.

May 27, 2019
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(Guest:   Note....you cannot have a   645     .....  there is no '6'   in base 5 )

ElectricPavlov  May 27, 2019
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re:  there is no '6' in base 5

.

Guest May 27, 2019
#2
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I think the answer to the third one is   17  base 10

base 5 would be  32

base 7 would be  23

May 27, 2019
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1)    554 in base b      5*(b^2) + 5(b) + 4

must equal   24b2 =    [2(b)+4) ]2

Eqaute the two of them:

5b^2 + 5b + 4 = 4b^2 +16b+16

b^2 -11b-12 = 0

(b-12)(b+1)  = 0        b = 12      (throw out the b = -1)

May 27, 2019
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Very nice, EP  !!!!   CPhill  May 27, 2019
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2)

12 sections  * x pages   + 11 pages   must be less than 100 and a multiple of 5    + 1

12(x) + 11 < 100       11 is already a multiple of 5     + 1

so we need 12x to be a multiple of 5       60 will work nicely    so x = 5    (the next multiple of 5  would be 120....too big)

12(5) + 11 = 71 pages     (assuming 2nd to last page = next to last page)

May 27, 2019
edited by ElectricPavlov  May 27, 2019
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#6
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You're welcome !   I hope you actually understand the answers and how they were derived..... ElectricPavlov  May 27, 2019