Reverse Cryptarithm

Logic Level 4

$\large \begin{array}{ccccc} & \color{#D61F06}{A} & \color{#3D99F6}{B} &\color{#ffbf00}{C} &\color{#20A900}{D}\\ - & \color{#20A900}{D} & \color{#ffbf00}{C} & \color{#3D99F6}{B} & \color{#D61F06}{A} \\ \hline & \color{#3D99F6}{B} & \color{#20A900}{D} & \color{#D61F06}{A} & \color{#ffbf00}{C} \end {array}$

The above shows a cryptarithm such that each letter represents a distinct digit.

Calculate the 4-digit integer, $\overline{BDAC}$ .

The answer is 6174.

1 solution

Worranat Pakornrat
Nov 27, 2015

1000A+100B+10C+D -(1000D+100C+10B+A) = 1000B+100D+10A+C

989A-1099D = 910B+91C = 91(10B+C)

989 = -12 (mod 91); 1099 = 7 (mod 91)

So 12A+7D = 0 (mod 91).

91= $13\times 7$ = $12\times 7$ +7.

So A=7; D=1.

Then 91(10B+C) = $989\times 7$ -1099 = 5824. 10B+C = 64; B=6; C=4.

Thus, the answer is 6174.

Can u explain the step following 989A-1099D = 910B+91C = 91(10B+C) clearly, I didn't understand how we got the values of A and D

Suneel Kumar - 5 years, 6 months ago

It's called modular operation. For example, when you divide 1099 with 91, you get a remainder of 7, so 1099 = 7 (mod 91). Then we can treat such congruence as one kind of equations on the same modulus used, 91 in this case.

Worranat Pakornrat - 5 years, 6 months ago

Do you know the speciality of that four digit number 6174 it's kaprekar number

Hemanth Koundinya - 5 years, 6 months ago

Oh, unfortunately I don't. What does it mean?

Worranat Pakornrat - 5 years, 6 months ago

Kaprekar is a mathematics teacher in a primary school in India during 20th century and

First consider any four digit nums except nums as 9999 8888 7777 etc
For eg : Consider 1234.
Now arrange it in descending order and subtract with it's ascending order 4321-1234=3087

Similarly 8730-0378=8352

8532-2358=6174

7641-1467=6174

Similarly you can do this for any four digit number except those on top.

Hemanth Koundinya - 5 years, 6 months ago

@Hemanth Koundinya – Nice!!! How would u guys know all these, I am of X standard but what is the min standard to get a grip on all these? And what's ur age

Suneel Kumar - 5 years, 6 months ago

@Suneel Kumar – My age is 18 I'm a b tech student in Karnataka.actually there is no minimum standard or age to know all these please keep on surfing about mathematical achievements by Indians that's all required

Hemanth Koundinya - 5 years, 6 months ago

@Hemanth Koundinya – Rolle's theorem is given by bhaskara charya 2 in 5th century

Hemanth Koundinya - 5 years, 6 months ago

Reverse Cryptarithm

The answer is 6174.

1 solution

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