Question

What is the remainder when 7^26 *5^ 83 is divided by 100 ?

Answer 1

Given:

The dividend is given by,   $\[{7^{26}} * {5^{83}}\]$.

The divisor is given as 100.

To Find:

We have to find the reminder for $\[\left( {{7^{26}} * {5^{83}}} \right) \div 100\]$.

Solution:

There are two ways to solve the problem,

1. Simplification
2. Find the pattern

For this question, we can choose the second method.

Since the divisor is 100, the reminder can be a two-digit number.

The first part is 7²⁶,

                               $\[\begin{array}{l}{7^1} = 7\\{7^2} = 49\\{7^3} = 343\\{7^4} = 2401\\{7^5} = 16807\\{7^6} = 117649\,.......\end{array}\]$

We can see the last two digits, i·e· 07, 49, 43, 01 is repeating in a cycle. The cycle repeats every four steps.  The power 26 is not a factor of 4 and leaves a reminder 2.

So, the last two digits of 7²⁶ are 49.

Similarly, we have to find the pattern for 5⁸³.

                                $\[\begin{array}{l}{5^1} = 5\\{5^2} = 25\\{5^3} = 125\\{5^4} = 625\\{5^5} = 3125\\{5^6} = 15625\,.......\end{array}\]$

From the above pattern, it is clear that whatever be the power for 5 the last two digits is 25 expect for 5¹.

So, the last two digits of 5⁸³ are 25.

∴ By multiplying the last two digits of 7²⁶ and 5⁸³, we get

                              $\[49 * 25 = 1225\]$

∴ When it is divided by 100, we get a reminder as 25.

Hence, the reminder for $\[\left( {{7^{26}} * {5^{83}}} \right) \div 100\]$ is 25.