Find Remainder of $40^{110}$ and $3^{1000}$ Divided by 37 and 26

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In summary, the formula for finding the remainder of a number divided by another number is: remainder = (dividend % divisor), where "%" represents the modulo operator. To find the remainder of $40^{110}$ divided by 37, we can use this formula and divide the result by 37 to find the remainder. The remainder when dividing by a number can indicate whether the dividend is evenly divisible by the divisor or not. The value of the remainder can change when dividing by different numbers and can be negative if the dividend is negative and the divisor is positive.
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Albert1
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(1)dividing $40^{110} \,\, by \,\, 37$

(2)dividing $3^{1000} \,\, by \,\, 26$


 
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  • #2
Albert said:
(1)dividing $40^{110} \,\, by \,\, 37$

(2)dividing $3^{1000} \,\, by \,\, 26$



(1)
we have
40 = 3 mod 37
so we need
$3^{110}$ mod 37
as 37 is prime so we have as per flt
$3^{36} = 1$ mod 37
so
$3^{110} = 3^{3*36+2} = (3^{36})^{3} * 3^2$ mod 37 = 9 mod 37 = 9 that is the ans

so $40^{110}$ devided by 37 remainder is 9

(2)
$3^{3} = 27 = 1$ mod 26
hence
$3^{999} = (3^3)^{333} = 1$ mod 26
so $3^{1000}$ mod 26 = 3 so remainder divided by 26 is 3
 

FAQ: Find Remainder of $40^{110}$ and $3^{1000}$ Divided by 37 and 26

What is the formula for finding the remainder of a number divided by another number?

The formula for finding the remainder of a number divided by another number is: remainder = (dividend % divisor), where "%" represents the modulo operator.

How do you find the remainder of $40^{110}$ divided by 37?

To find the remainder of $40^{110}$ divided by 37, we can use the formula above. First, we need to calculate $40^{110}$ using a calculator or by hand. Then, we can divide the result by 37 and find the remainder using the modulo operator. The answer will be the remainder of $40^{110}$ divided by 37.

What is the significance of the remainder when dividing by a number?

The remainder when dividing by a number can indicate whether the dividend is evenly divisible by the divisor or not. If the remainder is 0, then the dividend is divisible by the divisor. If the remainder is not 0, then the dividend is not divisible by the divisor.

How does the value of the remainder change when dividing by different numbers?

The value of the remainder can change when dividing by different numbers. For example, when dividing by a smaller number, the remainder will be smaller. When dividing by a larger number, the remainder will be larger. The value of the remainder is also affected by the values of the dividend and divisor.

Can the remainder be negative?

Yes, the remainder can be negative. This happens when the dividend is a negative number and the divisor is a positive number. In this case, the remainder will also be negative. However, in most cases, we use positive numbers for both the dividend and divisor, so the remainder will be positive.

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