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jarptica [38.1K]
3 years ago
5

Find the following Euler Totients using Euler’s Theorem, as explained on p.409 of the text (10 points each): a.ϕ(13) b.ϕ(81) c.ϕ

(100) d.ϕ(102)
Mathematics
1 answer:
ad-work [718]3 years ago
4 0

(a) \varphi(13)=12 since 13 is prime.

(b) 81=3^4, and there are 81/3 = 27 multiples of 3 between 1 and 81, which leaves 81 - 27 = 54 numbers between 1 and 81 that are coprime to 81, so \varphi(81)=54.

(c) 100=2^2\cdot5^2; there are 50 multiples of 2, and 20 multiples of 5, between 1 and 100; 10 of these are counted twice (the multiples of 2*5=10), so a total of 50 + 20 - 10 = 60 distinct numbers not coprime to 100, leaving us with \varphi(100)=100-60=40.

(d) 102=2\cdot3\cdot17; there are 51 multiples of 2, 34 multiples of 3, and 6 multiples of 17, between 1 and 102. Among these, we double-count 17 multiples of 2*3=6, 3 multiples of 2*17=34, and 2 multiples of 3*17=51; we also triple-count 1 number, 2*3*17=102. There are then 51 + 34 + 6 - (17 + 3 + 2) + 1 = 70 numbers between 1 and 102 that are not coprime to 102, and so \varphi(102)=102-70=32.

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s2008m [1.1K]

Answer:

Step-by-step explanation:

Unfortunately, your " 12×+9=10×1 " can be interpreted in more than one way.  The symbol " × " should be used ONLY to indicate multiplication.  Thus, " 12× " reads as ' 12x.'  Then we have:  12x + 9 = 10.  Unsure of what you meant by " 10×1​".  So I will treat your "10×1​" as simply '10.'

Then we have 12x + 9 = 10, which reduces to 12x = 1 if we subtract 9 from both sides.

12x = 1 yields  x = 1/12.

If this is not the result you expected, please go back to the original problem and share it as accurately as you can.

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Based on her recipe, Elena knows that 5 servings of granola have 1,750 calories. 1. if she eats 2 servings of granola, how many
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2 years ago
Can someone explain 5. please ?
GuDViN [60]

Both denominators are 4, so we can add the numerators to place over the common denominator. The numerators are -3 and -3, which add to -6. One way to think of negative numbers is to think of IOUs, which are a way of expressing debt in money. For instance, if you go into a store and buy a $10 item, but only have $3 in your pocket, then you would have to owe the owner $7. This can be represented with -7. If you repeat the process, then you'd have 7+7 = 14 in IOUs total. This would be represented with -14

In short, adding negative numbers is really the same as adding positives, but the final result is negative

So that's why -3+-3 turns into -6. We add the two threes like normal but then make the final result negative. All throughout this process, the denominator stays at 4.

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5 0
3 years ago
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