In a 3-digit number sum of its all three digits is less than 20 and the sum of hundredths place digit and ten's digit is equal to 10, but while reversing the order of the digits the new formed number is divided by 16 is equal to 58. Find the number
Answers
Step-by-step explanation:
A three-digit number which is being subtracted from another three-digit number consisting the same digit in reverse order gives 594. What is the minimum possible sum of these three-digit numbers?
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Answer: 808
Solution/Explanation:
In subtraction the number subtracted is termed the subtrahend , and the number from which the subtrahend is subtracted is called the minuend. In our problem, both the subtrahend and the minuend consist of three digits each. If zyx denotes the subtrahend, then by hypothesis, the minuend consists of the same digits as the subtrahend but in reverse order, that is minuend is xyz.
Their values in decimal system are:
Value of xyz = 100x + 10y + z
and the value of zyx = 100z + 10y + x
Since by definition xyz - zxy = 594, we write
100x + 10y + z - (100z + 10y + x) = 594
Or, 100x + 10y + z - 100z - 10y - x = 594
Or, 99x - 99z = 594 This gives
x - z = 6…………………………………………………………………………….(1)
The number xyz has to fulfill the following conditions.
1) The digit in hundred place can not be 0 for then xyz would reduce to a two digit number and we know the number has 3 digits.
2) The digit in unit place of xyz cann’t be zero because it would give zero in hundred place after reversing xyz and that would make zyx a two digit number which is not permissible.
3) xyz has to be greater than 594 so that after subtraction, the difference xyz-zyx is 594, a positive quantity.
4) As per the relation (1), the digit in unit place is to be less than the digit in hundred place and their difference should be 6, that is x-z should be 6.
5) The end digits can not be equal because then their difference (xyz - zyx) would be zero and we know the difference to be 594, a nonzero quantity.
6) There is no restriction on the middle digit (y) in the ten-place as it remains invariant when xyz → zyx .
7) In order that the sum (xyz + zyx) is minimum, the digit in ten- place should be zero.
The number that satisfies all the above conditions is 701. Reversing it, the changed number is 107.
And the minimum possible sum = 701+107 = 808
You can see from the three digit number 701 that x (digit in the hundred place) = 7, z (digit in unit place) = 1 and their difference x-z = 6 which agrees with that given in relation (1).
Also, the difference between the two numbers, xyz - zyx = 701 - 107 = 594, in agreement with the data given in the question .