## Mathematical reasoning based on factor analysis is the key to the solution

### The problem: 6th Hard number system question for CAT

The product of a two digit number and a second number formed by reversing the digits of the first is 2430. Find the lower number.

**Comment:** Solve this sixth hard number system question for CAT by mathematical reasoning based on factors multiples concept.

### Solution to the 6th hard number system question for CAT: Step by step Mathematical reasoning based on Factors multiples concept

Let the two numbers be (ab) and (ba) where $a$ is the tenth digit of the first and $b$ is the units digit of the first number.

Let us form the mathematical relationship from the problem description.

By **place value mechanism**, first number is expressed as $10a + b$ and the second number as $10 b + a$.

Their product is equal to 2430,

$(10a + b)(10 b + a)=2430$

This is a 2 degree equation in two variables impossible to solve mathematically by itself.

We have to find revealing patterns to simplify the relation further.

When you examine the number 2430 closely, you discover the distinctive single appearance of the factor 5 in it.

The single factor of 5 have to have its corresponding partner in the LHS in one of the two expressions in the product.

Examine the two expressions.

Both are of essentially same structure - one term with coefficient 10 and the other with unit coefficient.

One of the unit coefficient terms must contribute then the factor of 5 that would come out of the brackets to compensate the factor 5 in the RHS.

Continuing our analysis in the same path, it is also realized that both $a$ and $b$ cannot have a factor of 5. If it were so, the product on the LHS would have then a factor of 25 that is not there on the RHS.

Conclusion:Either $a$ or $b$ must have a factor of 5. It follows, as both are less than 10, theterm with factor of 5 must be 5 itself.

This is **factor analysis** by **factors and multiples concept.**

Let us assume $b=5$ as both $a$ and $b$ are symmetrically equivalent.

Substituting the value in the single equation we have,

$(10a + 5)(50 +a)=2430$,

Or, $(2a+1)(50+a)=486=2\times{3^5}$

We will again analyze factors on both sides of the equation.

**Observation:** The single factor of 2 on the RHS cannot be compensated by $(2a+1)$ on the LHS as it must be odd.

Conclusion 2:$a$ must be even.

**Observation:** Also the five numbers of factor 3 on the RHS forces $(2a+1)$ to comprise solely of factors of 3.

Conclusion:As $a$ is even and $(2a+1)$ must solely be comprising of factors of 3, $a=4$, and $2a+1=9=3^2$.

*The first number in the problem is then $45$ and the second $54$.*

*Lower of the two $45$ is your answer.*

### End note

The solution is easily visualized much earlier, but for driving home the power of mathematical reasoning based on irrefutable concepts, the whole logic has been put forth.

In many situations of hard problem solving, direct mathematical deduction is not easily visible but mathematical reasoning can easily break through.

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