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 Which digit should replace A in the units place of the number 987,654,32A so that the number is divisible by 6?

 Jun 2, 2018
 #1
avatar+4330 
+2

Divisible by 6, means divisible by 2 and 3.

 

Divisible by 2: Even units digit

 

Divisible by 3: Sum of digits must be divisible by 3.

 

We have 0,2,4,6,8 for the possibilities for being divisible by 2:

 

For the divisibility of 3, let's add the values up: 9+8+7+6+5+4+3+2=44

 

Since the divisibility of 2, means an even units digit, we have:

44+0=44

44+2=46

44+4=48

44+6=50

44+8=52

 

These values are divisible by 2 (for sure), but we need to check if they're divisible by 3.

 

44=4+4=8, not divisible by 3

46=4+6=10, not divisible by 3

48= 4+8=12, it is divisible by 3 (yes!)

50=5+0=5, not divisible by 3

52=5+2=7, not divisible by 3

 

We have one possible value, which is 48.

 

Thus, our answer is \(48-44=\boxed{4}\)

smileysmiley

.
 Jun 2, 2018
edited by tertre  Jun 2, 2018
 #2
avatar+107493 
+1

Thanks, tertre!!!....here's one more method

 

If the number is even and the sum of its digits is divisible by 3, it will be divisible by 6

Thus...we  can ignore any digit, or sums of digits in the number that are themselves divisible by 3

 

So...we can ignore the leading 9, the 87 that follows, the 6, the 54 that follows and the 3

 

So...we' are left  with  2A

 

This is only divible by 6, if A  is even and the sum of 2 + A is divisible by 3

 

Thus....2 + 4 = 6    is the only possibility.....so....A  = 4

 

 

 

cool cool cool

 Jun 2, 2018
edited by CPhill  Jun 2, 2018

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