Question

Find the two-digit number satisfying the following three properties
[1] The number is less than 20
[2] The sum of the digits of the number is 10
[3] The number is odd.

Answer 1

Hint: Assume that the digit at units place is x and the digit at tens place is y. Use the fact that the number is less than 20 to claim that$y=0,1$. Use the fact that since the number is a two-digit number and hence y cannot be 0. Hence find the value of y. Use the fact that the sum of the digits of the number is 10 to form an equation in x. Solve for x and hence find the value of x. Hence find the number satisfying the above properties. Verify your answer.

Complete step-by-step answer:
Let the number at units place be x, and the number at tens place be y.
Since the number is less than 20, we have
$10y+x < 20$
Since $x\ge 0$, we have
$10y < 20$
Dividing both sides by 10, we get
$y < 2$
Hence, we have y = 0 or y = 1.
Since the number is a two-digit number, the digit at tens place cannot be 0. Hence, we have y= 0 is rejected.
Hence, we have y =1.
Now, we know that the sum of the digits of the number is 10.
Hence, we have $x+1=10$
Subtracting 1 from both sides, we get
$x=9$
Hence, we have the number at units place is 9 and the number at tens place is 1.
Hence, the number is 19.

Note: Verification:
We can verify the correctness of our solution by checking that the number 19 satisfies all of the above properties.
[1] Clearly, 19 is less than 20. Hence the first property is satisfied.
[2] Since 19 is not divisible by 2, 19 is an odd number. Hence the second property is also satisfied.
[3] Sum of digits of 19 is 1+9 =10 . Hence the third property is also satisfied.
Hence our answer is verified to be correct.