Curious Numbers
Number Theory
Level
3
There are 3 distinct 9-digit numbers which have the following characteristics:
- The number is divisible by 9;
- If we remove the last digit from one number, the remaining number is divisible by 8;
- If we remove the last two digits from one number, the remaining number is divisible by 7;
- And so on until we remove the last 8 digits, in which case the remaining number is divisible by 1.
What is the sum of 3 numbers?
1324567892
1234567890
4243457615
1546513614
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Three numbers are separately divisible by 9, which means the total is also divisible by 9. Thus we can eliminate two choices. The remaining choices are 1234567890 and 1546513614. Let the correct answer be 'X'.
If we add the three numbers after eliminating the last one digit the sum should be divisible by 8. Let that sum be 'Y'. Then, 10Y also should be divisible by 8.
Case1: If X = 1234567890, then the nearest 10Y value will be 1234567840 (Y=123456784, multiple of 8, next multiple is 123456792, but then 10Y will be greater than X, which is not possible). Then (X-10Y) = 50.
Case2: If X = 1546513614, then the nearest 10Y value will be 1546513600 (Y=154651360, multiple of 8). Then (X-10Y) = 14.
As Y is obtained by adding the three numbers after removing the last one digit, 10Y will be such that the sum of three numbers after substituting the last digit of each number by 0. From this we can conclude that the maximum possible difference between X and 10Y will be 27, (9 x 3).
But in Case 1, it 50, so Case 1 is not possible. Thus the answer is 1546513614.
For checking the answer I have found out the three distinct 9 digit numbers also....They are.............
Number 1 = 522006885
Number 2 = 522852408
Number 3 = 501654321
Total = 1546513614.