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Find the remainder of \[\dfrac{{{{51}^{49}}}}{1}\].

Last updated date: 13th Jul 2024
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Hint: Recall the definition of a remainder and the special properties when a number is divided by 1. We can also proceed with the equation \[n = q.m + r\], to determine the remainder. Don’t be conceived away with the magnitude of the dividend.

Complete step-by-step answer:

 Remainder is the integer left over after dividing one integer by another to produce an integral quotient.
Not all integers are exactly divisible by the other. When they are not exactly divisible, they can be written in the form:
\[n = q.m + r..........(1)\]
where n is the dividend, m is the divisor, q is the quotient and r is the remainder.
In the given question, we need to find the remainder when \[{51^{49}}\] is divided by 1.
Hence, \[{51^{49}}\] is the dividend, 1 is the divisor.
We can multiply 1 by \[{51^{49}}\] to obtain \[{51^{49}}\].
Hence, the quotient is \[{51^{49}}\].
Using formula (1), we can find the remainder as follows:
\[{51^{49}} = {1.51^{49}} + r\]
1 multiplied with any number is the number itself, hence, we have:
\[{51^{49}} = {51^{49}} + r\]
We cancel \[{51^{49}}\] on both sides of the equation to obtain:
\[r = 0\]
Hence, the remainder is 0.
Therefore, the correct answer is 0.

Note: You can directly use the special property of 1, that is, when any number is divided by 1, the remainder is zero and the quotient is the number itself. You might make an error in the concept and write the answer as 1, which is wrong.