Target Olympiad
1. Find the values of A and B if the given number 7A5798B8 is divisible by 33.
Answers
Question :- Find the values of A and B if the given number 7A5798B8 is divisible by 33.
Solution :-
Since , 7A5798B8 is divisible by 33 = 3 * 11 . Then it must be divisible by both 3 and 11.
we know that,
- if sum of all digits is divisible by 3, then the number also divisible by 3.
- A number to be divisible by 11 , the difference between the sum of the odd numbered digits (1st, 3rd, 5th...) and the sum of the even numbered digits (2nd, 4th...) must be divisible by 11.
So,
By Divisibility Rule of 3 we get :-
→ (7 + A + 5 + 7 + 9 + 8 + B + 8) ÷ 3
→ (44 + A + B) ÷ 3
So, Possible values of (A + B) are :-
- 45 - 44 = 1
- 48 - 44 = 4
- 51 - 44 = 7
- 54 - 44 = 10
- 57 - 44 = 13
- 60 - 44 = 16 .
By Divisibility Rule of 11 now, we get :-
→ { (7 + 5 + 9 + B) - (A + 7 + 8 + 8) } = 0, 11, 22, 33 ____
→ (21 + B) - (23 + A) = 0, 11, 22, 33 ____
→ B - A - 2 = 0, 11, 22, 33 __________
→ B - A = 2 , 13, 24 , 35 _________
→ B - A = Only 2 Possible.
From Both Conclusions , we get ,
→ A + B ≠ 1 , as B - A = 2 .
→ A + B = 4 and B - A = 2 => A = 1, B = 3 .
→ A + B = 7 and B - A = 2 => Not Possible.
→ A + B = 10 and B - A = 2 => A = 4 , B = 6 .
→ A + B = 13 and B - A = 2 => Not Possible.
→ A + B = 16 and B - A = 2 => A = 7, B = 9 .
Hence, the values of A and B are :- (1,3) , (4,6) and (7,9). (Ans.)
[ Excellent Question. ]