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Alexandra [31]
2 years ago
12

Prove that if a is a natural number, then there exist two unequal natural numbers k and l for which ak−al is divisible by 10.

Mathematics
2 answers:
EastWind [94]2 years ago
8 0
Assume a is not divisible by 10. (otherwise the problem is trivial). 
<span>Define R(m) to be the remainder of a^m when divided by 10. </span>
<span>R can take on one of 9 possible values, namely, 1,2,...,9. </span>
<span>Now, consider R(1),R(2),......R(10). At least 2 of them must have the sames value (by the Pigeonhole Principle), say R(i) = R(j) ( j>i ) </span>
<span>Then, a^j - a^i is divisible by 10.</span>
Elodia [21]2 years ago
8 0
Divisible by 10.........................................
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QveST [7]

Terrence is not correct.

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4 0
2 years ago
Consider f and c below. f(x, y) = (3 + 4xy2)i + 4x2yj, c is the arc of the hyperbola y = 1/x from (1, 1) to 3, 1 3 (a) find a fu
irga5000 [103]
We want to find a scalar function f(x,y) such that \nabla f(x,y)=\mathbf f(x,y)=\dfrac{\partial f}{\partial x}\,\mathbf i+\dfrac{\partial f}{\partial y}\,\mathbf j.
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4 0
3 years ago
Plz help<br> Don’t know which one it is?
ddd [48]

Answer:

 First option  6^\frac{1}{6} is the correct answer.

Step-by-step explanation:

Let us consider each part, we have number 6 inside all the roots.

First we have to find cube root of 6.

                    \sqrt[3]{6} =6^\frac{1}{3}

Now we have to find square root of this value.

                   \sqrt{6^\frac{1}{3}  } =(6^\frac{1}{3} )^\frac{1}{2} =6^\frac{1}{6}

So, first option  6^\frac{1}{6} is the correct answer.

3 0
3 years ago
Joe Hernandez has inherited $25,000 and wishes to purchase an annuity that will provide him with a steady income over the next 1
mote1985 [20]

Answer:

p=2982

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25000 = P[1-(1+0.06)-12/0.06]

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8 0
3 years ago
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skelet666 [1.2K]
A is correct.

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3 years ago
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