Question

If we add the square of the digit in the tens place of a positive two digit number to the product of the digits of that number we shall get 52 and if we add the square of the digit in the units place to the same product of the digits we shall get 117. Find the two digit number.

Answer 1

Hint:
We will use the conditions given in the problem itself. We will assume the two digits be a and b. then apply the condition on them. We will get an equation that leads towards the solution. So let’s solve it!

Complete step by step solution:
Let a be the digit on the unit's place and b on ten’s place.
From the first condition, if we add the square of the digit in the tens place of a positive two digit number to the product of the digits of that number we shall get 52.
the square the digit in the tens place of a positive two digit number→ \[{b^2}\]
product of the digits of that number→ \[ab\]
if we add ,we shall get 52
\[{b^2} + ab = 52..... \to 1\]
From second condition, if we add the square of the digit in the units place to the same product of the digits we shall get 117
the square of the digit in the units place→ \[{a^2}\]
same product of the digits→ \[ab\]
if we add ,we shall get 117
\[{a^2} + ab = 117..... \to 2\]
Now perform \[2 - 1\] we get,
\[{a^2} + ab - {b^2} - ab = 117 - 52\]
\[ \Rightarrow {a^2} - {b^2} = 65\]
Now we know the identity that \[ \Rightarrow {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
Thus applying it here only,
\[
   \Rightarrow \left( {a + b} \right)\left( {a - b} \right) = 65 \\
   \Rightarrow \left( {a + b} \right)\left( {a - b} \right) = 13 \times 5 \\
\]
Now we have to find two numbers a and b such that their sum is 13 and difference is 5.
Got it! The numbers are 9 and 4.
Because, 9+4=13 and 9-4=5
Thus \[\left( {a + b} \right) = 9 + 4 = 13\] and \[\left( {a - b} \right) = 9 - 4 = 5\]

Hence the two digit number is 49.

Note:
Here students should note that a is the bigger number and b is smaller (so a is 9 and b is 4) but a is on units place and b is on tens place. They might get confused and can write 94 instead of 49. Remember!