Respuesta :
The least positive integer n is 8 such that 10^n is expressed as the product of any two positive integers.
What is a positive integer?
- The numbers used for counting and sorting in mathematics are known as natural numbers.
- Cardinal numbers are those that are used to count, and ordinal numbers are those that are used to order.
- A whole number higher than zero is what constitutes a positive integer. Every counting number is contained in the set of positive integers (that is, the natural numbers)
How is it calculated?
10n has factors 2 and 5.
For n=1, [tex]2^{1}[/tex]=2, [tex]5^{1}[/tex]=5
For n=2, [tex]2^{2}[/tex] =4 [tex]5^{2}[/tex]=25
For n=3, [tex]2^{3}[/tex]=8, [tex]5^{3}[/tex]=125
For n=8 [tex]2^{8}[/tex]=256 [tex]5^{8}[/tex]=390625
Here, [tex]5^{8}[/tex] contains the zero then n=8.
The least positive integer n is 8 such that 10^n is expressed as the product of any two positive integers.
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The least positive integer nn such that no matter 10 n is expressed as the product of any two positive integers is n = 8
Calculating the problem:
A factor of $10n will end in a 0 if its prime factorization contains 2 and 5. In this way, we have left to consider the situation when the two elements have the 2s and the 5s isolated, so we want to find the primary force of 2 or 5 that contains a 0.
For n = 1 2 = 2¹ , 5₁ = 5
n = 2 2² = 4 , 5 ² =25
n = 3 2³ = 8 , 5³ = 125
and so on, until,
n = 8 2⁸ = 256 5⁸ = 390625
We see that 5⁸ contains the first zero, so n = 8
In math, what is a prime factorization?
Writing all numbers as the product of primes is known as prime factorization. Any number can be prime factorized in one of two ways:
Division strategy - In this technique, the given number is separated by the littlest indivisible number what partitions it totally. The quotient is then divided once more by the smallest prime number. The process continues until the quotient reaches 1. After that, the divisors of all prime factors are multiplied.
The given number is placed on top of the factor tree in the factor tree method. The corresponding pairs of factors are then represented as the tree's branches. The composite factors are then factorized once more and written down as the subsequent branches. This procedure is repeated until all the composite factors' prime factors are obtained.
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