if 5478A87B is divisibke by 66, then what is the value of A and B respectivel
Answers
A = 5 and B = 4 and the complete number is 54785874
Given data:
A number 5478A87B is divisible by 66.
To find:
The values of A and B.
Solution:
As 5478A87B is divisible by 66, it is also divisible by 2, 3, 11 (prime factors of 66)
Divisibility by 2 rule:
As the number is divided by 2 too, the value of B can be one of the digits 0, 2, 4, 6, 8, as the last number will always be even if it is divided by 2.
Divisibility by 11 rule:
Sum of digits at odd places - Sum of digits at even places=0 or 11
To calculate the small value, let us equate it to 0.
So,
(5 + 7 + A + 7) - (4 + 8 + 8 + B) = 0
19 + A - 20 - B = 0
A - B - 1 = 0
A - B = 1
A = B + 1 ------------------------- ( 1 )
Divisibility by 3 rule:
Sum of digits should be divisible by 3
(5 + 4 + 7 + 8 + A + 8 + 7 + B) = 3n
39 + A + B = 3n
From equation 1,
39 + B + 1 + B = 3n
40 + 2B = 3n
Around 40+, the factors of 3 are: 42, 45, 48
Starting from the least one, let us check one on one till we get B an even number following the divisibility by 2 rule.
40 + 2B = 42
2B = 2B = 1 [incorrect]
40 + 2B = 45
2B = 5B = 5/2 [incorrect]
40 + 2B = 48
2B = 8B = 4 [CORRECT!]
Then, A = B + 1 = 4 + 1 = 5
Therefore, A = 5 and B = 4 and the complete number is 54785874
Verification:
66 divides 54785874 by 830089 times.
2 divides 54785874 by 27392937 times.
3 divides 54785874 by 18261958 times.
11 divides 54785874 by 4980534 times.
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