## Approach strategically to discover the key patterns

### The problem: 2nd Hard number system question for CAT

What two digit number is less than the sum of squares of its digits by 11 and exceeds their doubled product by 5?

**Hint:** For quick solution of the 2nd hard number system question for CAT, discover the breakthrough information from the given resources.

### Solution to the 2nd hard number system question for CAT: Deriving key relations by discovering key patterns

By place value mechanism applied on the two given statements, where $x$ is the ten's digit and $y$ is the unit's digit of the number $N$,

$N=10x+y=x^2+y^2-11$ ....................(1)

$N=10x+y=2xy+5$ ............................(2)

Can we derive any special relation between the two digits from any of the equations independently?

We avoid the first equation with squares and no visible usable patterns.

But with specific intent when we examine equation 2, it is easy to discover the possibility of factorization of this second equation. The hint is taken from the common factor 2 in terms $2xy$ and $10x$,

$10x+y=2xy+5$,

Or, $y-5=2x(y-5)$,

Or, $(y-5)(2x-1)=0$.

This implies, either $(y-5)=0$ or $(2x-1)=0$.

If $(2x-1)=0$, $x=\frac{1}{2}$ that is an invalid value of ten's digit $x$. Then,

$(2x-1) \neq 0$.

So, $y-5=0$,

Or, $y=5$.

And this is an *important and quick breakthrough.*

Exhausting the exploitation of individual equations, in the next step, **two equations are considered together** to find a **second useful pattern** that we can reveal by a simple operation.

Again with specific intent, is easy to see that the fragment expressions $x^2+y^2$ in the first equation and $2xy$ in the second can give us the value of $(x-y)$.

Subtract equation 2 from equation 1,

$(x-y)^2=16$,

Or, $x-y=\pm 4$.

*What can be the possible values of $x$ and $y$ with these two key information available?*

The only two possibilities are,

$15$, and, $95$.

Note: First ten's digit value is fixed at $5$ then two ways of its differing from ten's digit by 4 gives the two values possible.

Let's verify.

#### Verification of the values against given conditions

**For 15, and Equation 1:** sum of squares of the digits is **26** that **exceeds** the number by **11**. Equation satisfied.

**For 15, and Equation 2:** Twice the product of the digits is **10** that is **less than** the number by **5**. Equation satisfied.

**For 95, and Equation 1:** Sum of squares of the digits is **106** that **exceeds** the number by **11**. Equation satisfied.

**For 95, and Equation 2:** Twice the product of the digits is **90** that is less than the number by **5**. Equation satisfied.

**Answer:** Two possible values of the number are 15 and 95.

**Concepts and techniques used:** Strategy of dealing with less complex individual equation first and the two equation together second to discover key patterns and derive key relations -- Key pattern discovery -- Key information derivation -- Enumeration based on constraints -- Systematic problem solving.

This is systematic, confident and quick problem solving using suitable concepts and techniques at each step, in short, **systematic problem solving**.

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