Question

Find the sum of smallest 3 digits even number and largest 3 digits odd number formed using the digits 6,2 and 3?
[a] 859
[b] 958
[c] 598
[d] 985

Answer 1

Hint: An even number is a number which on division by 2 gives remainder zero, e.g. 4 6,12,14 etc. An odd number is a number which on division by 2 leaves remainder 1, for example, 3, 7 ,15, 1999 etc.

Complete step-by-step answer:

When making a number as large as possible, we keep the larger digits on the left-hand side and smaller digits on the right-hand side. Consider the case in which we are asked to make a number as large as possible using the digits 3, 5, 9 we keep the larger digit 9 on the left-hand side followed by 5 followed by 3. Hence, we get the number 953, which is the largest number which can be made using these digits. Now consider the case in which we are asked to create the largest number using the digits 2,5,7 and 9 so that it is also an even number. In this case we will keep the smallest even number digit on the right-hand side. In this case the smallest even digit number 2 will be kept on the right hand side and the remaining digits will be used to create the largest number; this number will then be appended to that last digit thus forming the largest even number. Hence in this case the largest even number will be 9752. If you are asked to create the largest part number using digits 3, 5, 7, 9, 2 and 6. In this case we will keep the smallest odd digit on the right hand side and create the largest number using the rest of the digits then we will append this new number formed with the digit and hence the number formed will be the largest odd number. Hence the largest number formed will be 976523.
When making a number as small as possible using the provided digits we keep the larger digits on the right hand side. Thus the smallest number which can be formed using the digits 3,5 and 9 we will keep the smallest digit 3 on the left hand side followed by 5 followed by 9. Hence the number formed will be 359.
If asked to create the smallest even number using some digits, then in that case we will keep the largest even digit on the right hand side and create the smallest number using the rest of the digits then we will append this newly formed with the digit and hence the number we get will be the smallest even number. Consider the case in which we are asked to create the smallest even number using the digits 2, 3, 5, 7, 8 and 9. In this case the largest even number is 8 so we will keep it apart and create the smallest number using the rest of the digits which will be 23579 and append this number to 8 and hence we will get the number 235798 which will be the smallest even number that can be formed using the digits. To create the smallest number which is odd using some digits then in that case we will keep the largest odd number on the right hand side and create the smallest number using the rest of the digits and append this new number with the remaining digit and hence, we will get the smallest odd number.
Following the above-mentioned procedure, the smallest even number that can be formed using digits 6,2 and 3 is 236 and the largest odd number which can be formed using these digits will be 623.
Hence the sum of smallest three digit even number formed using the digits 6,2,3 and largest odd number formed using these digits is 236+623 =859
Hence option [a] is correct.
Note: Even numbers have even digits at one’s place whereas odd numbers have odd digits at one’s place.
Consider the number ${{a}_{1}}{{a}_{2}}\ldots {{a}_{n}}$then ${{a}_{1}}{{a}_{2}}\ldots {{a}_{n}}=10\times {{a}_{1}}{{a}_{2}}\ldots {{a}_{n-1}}+{{a}_{n}}$
Since $10\times {{a}_{1}}{{a}_{2}}\ldots {{a}_{n-1}}$ is divisible by 2 because 10 is divisible by 2 we get
$2|{{a}_{1}}{{a}_{2}}\ldots {{a}_{n}}\Leftrightarrow 2|{{a}_{n}}$
Hence a number is divisible by 2 if and only if the digit at one’s place is divisible by 2.
Hence even numbers have even digits at one’s place whereas odd numbers have odd digits at one’s place.