SOLUTION: Find the value of the digit A if the 5-digit number 12A3B is divisible by both 4 and 9, and A =/ B.

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Question 114750: Find the value of the digit A if the 5-digit number 12A3B is divisible by both 4 and 9, and A =/ B.
Answer by MathLover1(20849) About Me  (Show Source):
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12A3B
Apply following Divisibility rules:
Dividing By 4: A number is divisible by 4 if the last%85+2-digits=3B+ are divisible by 4.
So, digits could be an even number:
B=2 or B=6
Let’s try B=2:
3B=+32
32%2F4=8………… divisible by 4…..
Let’s try B=6:
3B=+36
36%2F4=9……………… divisible by 4…..



Apply now rule Dividing By 9: A number is divisible by+9 if the sum+_of_+the+_digits is divisible by 9; so, sum+=+1%2B2%2BA%2B3%2BBhas to be divisible by 9.
Now check which B would be right solution:
12A3B
if B=2, we have 12A32

Since the sum of known digits is 1%2B2%2B+A+%2B3%2B2+=+8+%2BA, we need a such of number A to get the sum of the digits as a multiple of 9 in order
to get a number 12A32 divisible by 9.

The first number added to this8 is 1; so,
A=1
if B=6, we have 12A36
we see that 1%2B2%2BA%2B3%2B6=12%2BA, so if B=6 then+A=6 too because 12%2B6=18 and 18+is divisible by 9; but, to have A=B is not allowed.
So, if A=1andB=2 our number is: 12132
check

the last%85+2-digits=32+…. 32 is divisible by 4
the sum+_of_+the+_digits+=+1%2B2%2B1%2B3%2B2=9………9 is divisible by 9
the 12132:
12132%2F4=3033
12132%2F9=1348