Question

The number 3^13 - 3^10 is divisible by a. 3,13,5 b. 3,10 c. 2,3,13 d. 2,3,10 solve

Answer 1

ANSWER:

• Option c) 2, 3, 13 is correct.

GIVEN:

• $\sf 3^{13}- 3^{10}$

TO FIND:

• The divisibilty of the given number.

EXPLANATION:

$\sf 3^{13}- 3^{10}$

$\sf \underline{Take \ 3^{10}\ as\ common}$

$\sf3^{10}( {3}^{3} - 1)$

$\sf3^{10}( 27 - 1)$

$\sf3^{10}( 26)$

a) 3, 13, 5

➳ Divide by 3

$\sf \dfrac{ 3^{10}( 26)}{3} =26 \times 3^{9}$

➳ It is divisible by 3

➳ Divide by 13

$\sf \dfrac{ 3^{10}( 26)}{13} =2 \times 3^{10}$

➳ It is divisible by 13

➳ Divide by 5

$\sf \dfrac{ 3^{10}( 26)}{5} =\dfrac{ 243 \times 243 \times 26}{5}$

As the numbers doesn't end with zero or five it is not divisible by 5

Option a) is incorrect.

b)3, 10

◈ Divide by 3

$\sf \dfrac{ 3^{10}( 26)}{3} =26 \times 3^{9}$

◈ It is divisible by 3

◈ Divide by 10

$\sf \dfrac{ 3^{10}( 26)}{10} =\dfrac{ 243 \times 242 \times 26}{10}$

As the numbers doesn't end with zero it is not divisible by 10.

Option b) is incorrect.

c)2, 3, 13

➳ Divide by 2

$\sf \dfrac{ 3^{10}( 26)}{2} =13 \times 3^{10}$

➳ It is divisible by 2.

➳ Divide by 3

$\sf \dfrac{ 3^{10}( 26)}{3} =26 \times 3^{9}$

➳ It is divisible by 3

➳ Divide by 13

$\sf \dfrac{ 3^{10}( 26)}{13} =2 \times 3^{10}$

➳ It is divisible by 13

Option c) is correct.

d) 2, 3, 10

◈ Divide by 2

$\sf \dfrac{ 3^{10}( 26)}{2} =13 \times 3^{10}$

◈ It is divisible by 2.

◈ Divide by 3

$\sf \dfrac{ 3^{10}( 26)}{3} =26 \times 3^{9}$

◈ It is divisible by 3

◈ Divide by 10

$\sf \dfrac{ 3^{10}( 26)}{10} =\dfrac{ 243 \times 243 \times 26}{10}$

As the numbers doesn't end with zero it is not divisible by 10.

Option d) is incorrect.