**I**. Consider the following equation where and represent two relatively prime integers

(a) Verify that the ordered pair is a solution of Eq.

(b) Use part (a) to find a parametric solution of Eq.

(c) Derive an integer such that and

**II**. We want to encode a two-letter word by using the following procedure

STEP 1 Each letter is replaced by an integer according to the table below

A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
K |
L |
M |
N |
O |
P |
Q |
R |
S |
T |
U |
V |
W |
X |
Y |
Z |

0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
25 |

We obtain an ordered pair where corresponds to the first letter of the word and corresponds to the second letter of the word.

STEP 2 is transformed into such that

with and

STEP 3 is transformed into a two-letter word by using the correspondence table given in STEP 1

Example:

plain text encoded text

1. Encode the word

2. We want to determine the decoding procedure:

(a) Show that if verifies the system of equations then it verifies the system of equations

(b) Use part (I) to show that if verifies the system of equations then it verifies the system of equations

(c) Show that if verifies the system of equations then it verifies the system of equations

(d) Decode the word

Source: Baccalauréat Général, Série Scientifique, Session Avril 2012, Pondichéry, http://www.ilemaths.net

**SOLUTION**

**I**. (a) Verify that the ordered pair is a solution of the equation

is a solution of the equation

(b) Use part (a) to find a parametric solution of the equation

Consider the system of equations

Multiply the second equation by and add the equations

——————————–

divides

divides because and are relatively prime

for some integer

Substitute the value of into Eq.

Simplify

A parametric solution of Eq. is

(c) Derive an integer such that and

If then there exists an integer such that

Simplify

From part (b)

for

The multiplicative inverse of is .

**II**. 1. Encode the word

STEP 1 The plain text corresponds to

STEP 2 Transform into by substituting the values of and into the system of equations

Simplify

Similarly

Simplify

STEP 3 From the table corresponds to the encoded text .

2. (a) Show that if verifies the system of equations then it verifies the system of equations

Given that verifies the system of equations

Substitute the value of and from into the right hand sides of

and

Similarly

and

If verifies the system of equations then it verifies the system of equations .

(b) Use part (I) to show that if verifies the system of equations then it verifies the system of equations

Given that verifies the system of equations

Multiply both sides of the system of equations by , the multiplicative inverse of

Similarly

and

If verifies the system of equations then it verifies the system of equations .

(c) Show that if verifies the system of equations then it verifies the system of equations

Given that verifies the system of equations

Substitute the values of and from into the right hand sides of

and

Similarly

and

If verifies the system of equations then it verifies the system of equations .

(d) Decode the word

The encoded text corresponds to

Substitute the values of and into the system of equations

Similarly

The plain text is .

**Answer**: Given in solution.