Question 1134705
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            The problem in the post is formulated by the very curved and twisted way,  using outdated and antiquated language.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In my opinion, &nbsp;the correct &nbsp;(adequate; &nbsp;straightforward and mathematically correct) &nbsp;formulation is &nbsp;<U>AS &nbsp;FOLLOWS</U>:


<pre>
            Find integer non-negative numbers "b" such that

            (5b + 6) = 0  mod (b+6).
</pre>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So, &nbsp;the problem is about solving congruences.



<U>Solution</U>


<pre>
    5b + 6 = 5(b+6) - 30 + 6,

    5b + 6 = 5*(b+6) - 24.


Therefore,


    5b + 6 = 0  mod (b+6)      (1)

if and only if

    -24 = 0  mod (b+6),


which is equivalent to

    24 = 0  mod (b+6).         (2)


From (2),  you easily find  b = 0, 2, 6, 18.


<U>Check</U>.


    a)  b = 0.  Then  5b + 6 = 5*0 + 6 = 0 + 6 = 6. From the other side,  b+6 = 6,  and  6 = 0 mod 6.   ! correct !


    b)  b = 2.  Then  5b + 6 = 5*2 + 6 = 10 + 6 = 16. From the other side,  b+6 = 8,  and  16 = 0 mod 8.   ! correct !


    c)  b = 6.  Then  5b + 6 = 5*6 + 6 = 30 + 6 = 36.  From the other side,  b+6 = 12,  and  36 = 0 mod 12.   ! Correct !


    d)  b = 18.  Then  5b + 6 = 5*18 + 6 = 90 + 6 = 96.  From the other side,  b+6 = 24,  and  96 = 0 mod 24.   ! Correct !


So, the problems has 4 solutions  b = 0, 2, 6 and 18.      <U>ANSWER</U>
</pre>

Solved.


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The formulations and wordings as in the original post, are used now in sections and circles of amateurs of the OLD ENGLISH only.


In professional Math it is not in use just at least 200 years,
and in the School Math it is not in use more than 100 years.


It is some "before-Newton's era" language . . .