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  • If the integers m and n are chosen at random between 1 and 100 then the probability that a number of the form <img alt="\mathrm{7^m+7^n}" src="https://entrancecorner.oncodecogs.com/gif.la

If the integers m and n are chosen at random between 1 and 100 then the probability that a number of the form \mathrm{7^m+7^n} is divisible by 5 is

Option: 1

\frac{1}{5}


Option: 2

\frac{1}{7}


Option: 3

\frac{1}{4}


Option: 4

\frac{1}{49}


Answers (1)

We know \mathrm{7^k, k \in N}, has 1,3,9,7 at the units place for \mathrm{k=4 p, 4 p-1, 4 p-2,4 p-3} respectively, where \mathrm{p=1,2,3, \ldots}.
Clearly, \mathrm{7^m+7^n} will be divisible by 5 if \mathrm{7^m} has 3 or 7 in the units place and \mathrm{7^n} has 7 or 3 in the units place or \mathrm{7^m} has 1 or 9 in the units place and \mathrm{7^n} has 9 or 1 in the units place.

\therefore for any choice of m, n the digit in the units place of \mathrm{7^m+7^n} is 2,4,6,0 or 8 . It is divisible by 5 only when this digit is 0.

\mathrm{\therefore \text { the required probability }=\frac{1}{5} \text {. }}

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Kshitij

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