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Among all 6-digit multiples of 22 of the form 5d5,77e where d and e are digits, what is the maximum value of d?

 Jul 15, 2022
 #1
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d==8,    e==2

 

585772 mod 22 ==0

 Jul 15, 2022
 #2
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Among all 6-digit multiples of 22 of the form 5d5,77e where d and e are digits, what is the maximum value of d?

 

5d577e = 0  (mod22)

5d577e = 505770 + 10000d + e

 

505770 + 10000d + e (mod22)

 

505770 (mod22) == 12 (mod22)

10000 (mod22) == 12 (mod22)

so

0== 505770 + 10000d + e (mod22)

== 12+12d+e  (mod22)

== -10(d+1) +e (mod22)

 

-10(d+1)

-10              

-20  

-30 +8 = -22         d=2, e=8                525778 (mod22) = 0

-40  

-50 + 6 = -44         d=4, e=6               545776 (mod22) = 0

-60

-70 + 4 = -66         d=6, e= 4              565774 (mod22) = 0

-80

-90 + 2 = -88        d=8, e= 2              585772 (mod22) = 0

-100

 

they are all the pairs that work.

 Jul 17, 2022

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