Find the remainder when $x = 5^{25}$ is divided by $24.$

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Solution

The correct option is A $5$
$(5)^{25}$
$(5) (5)^{24}$
$= 5 (25)^{12}$
$= 5 (1 + 24)^{12}$
$= 5 (1 +^{12} C_{1} 24^{1} +^{12} C_{2} 24^{2} + . . .)$
$= 5 (1 + 24 K)$ ...(where K is a positive natural number).
$= 5 + 120 K$
Dividing by $24$ we get
$\frac{5}{24} + 5 K$
Hence remainder will be $5$

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Find the remainder when x=525 is divided by 24.

The correct option is A 5(5)25(5)(5)24=5(25)12=5(1+24)12=5(1+12C1241+12C2242+...)=5(1+24K) ...(where K is a positive natural number).=5+120KDividing by 24 we get524+5KHence remainder will be 5

Find the remainder when $x = 5^{25}$ is divided by $24.$

The correct option is A $5$
$(5)^{25}$
$(5) (5)^{24}$
$= 5 (25)^{12}$
$= 5 (1 + 24)^{12}$
$= 5 (1 +^{12} C_{1} 24^{1} +^{12} C_{2} 24^{2} + . . .)$
$= 5 (1 + 24 K)$ ...(where K is a positive natural number).
$= 5 + 120 K$
Dividing by $24$ we get
$\frac{5}{24} + 5 K$
Hence remainder will be $5$