Solve 5 x ≡ 4 ( mod 6 ) .

We know that, Two integers a and b are congruence modulo n , if they differ by an integer multiple of n . This can also be written as a ≡ b ( mod n ) . Given, 5 x ≡ 4 ( mod 6 ) 5 x - 4 = 6 n , for some integer n 5 x = 6 n + 4 x = 6 n + 4 5 When we put n = 1 , 6 , 11 , … . then 6 n + 4 is divisible by 5 . x = 6 1 + 4 5 = 2 x = 6 6 + 4 5 = 8 Therefore, the solutions are 2 , 8 , 14 , …

Solve 3 x - 2 ≡ 0 ( mod 11 ) .

We know that, Two integers a and b are congruence modulo n , if they differ by an integer multiple of n . This can also be written as a ≡ b ( mod n ) . Given, 3 x - 2 ≡ 0 ( mod 11 ) Then, 3 x - 2 ≡ 0 ( mod 11 ) ⇒ 3 x - 2 = 11 n , for some integer n ⇒ 3 x = 11 n + 2 ⇒ x = 11 n + 2 3 , where n = 2 , 5 , 8 , … ⇒ x = 11 × 2 + 2 3 = 8 ∴ x = 11 × 5 + 2 3 = 55 + 2 3 = 57 3 = 19 Also, x = 11 × 8 + 2 3 = 88 + 2 3 = 90 3 = 30 ∴ x = 8 , 19 , 30 , …

Prove that 2 n + 6 × 9 n is always divisible by 7 for any positive integer n .

Two integers a and b are congruence modulo n , if they differ by an integer multiple of n . This can also be written as a ≡ b ( mod n ) . We know, 9 = 2 ( mod 7 ) ⇒ 9 n ≡ 2 n ( mod 7 ) and 2 n ≡ 2 n ( mod 7 ) Given, 2 n + 6 × 9 n ≡ 2 n ( mod 7 ) + 6 2 n ( mod 7 ) = 2 n ( mod 7 ) + 6 × 2 n ( mod 7 ) = 7 × 2 n ( mod 7 ) Hence, 2 n + 6 × 9 n is always divisible for any positive integer n .

E M B I B E

Mathematics Standard 10>Chapter 2 - Numbers And Sequences>Exercise 2.3>Q 2

HARD

10th Tamil Nadu Board

IMPORTANT

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If $x$ is congruent to $13$ modulo $17$ , then $7 x - 3$ is congruent to which number modulo $17$ ?

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Important Questions on Numbers And Sequences

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 2 - Numbers And Sequences>Exercise 2.3>Q 3

Solve

5 x \equiv 4 (\mod 6)

.

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 2 - Numbers And Sequences>Exercise 2.3>Q 4

Solve

3 x - 2 \equiv 0 (\mod 11)

.

MEDIUM

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 2 - Numbers And Sequences>Exercise 2.3>Q 5

If the time

100

hours after

7 a . m .

is

k a . m .

then find the value of

k

.

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 2 - Numbers And Sequences>Exercise 2.3>Q 6

If the time

15

hours before

11 pm

is

k p . m .

, then find the value of

k

.

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 2 - Numbers And Sequences>Exercise 2.3>Q 7

Today is Tuesday. My uncle will come after

45

days. In which day my uncle will be coming?

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 2 - Numbers And Sequences>Exercise 2.3>Q 8

Prove that

2^{n} + 6 \times 9 n

is always divisible by

7

for any positive integer

n

.

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 2 - Numbers And Sequences>Exercise 2.3>Q 9

Find the remainder, when

2^{81}

is divided by

17

.

HARD

10th Tamil Nadu Board

IMPORTANT

Mathematics Standard 10>Chapter 2 - Numbers And Sequences>Exercise 2.3>Q 10

The duration of flight travel from Chennai to London through British Airlines is approximately

11

hours. The aeroplane begins its journey on Sunday at

23 : 30

hours. If the time at Chennai is four and a half hours ahead to that of London's time, then find the time in London, when will the flight lands at London Airport on Monday. If the answer is