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Let n be a positive integer and prove that the number 2 Λ 2 Λ n + 2 Λ 2 Λ (n-1) + 1 has at least n different prime factors The formula is to the power of 2 (n power of 2), plus the power of 2 (n-1 power of 2), plus 1
Let a (n) = 2 ^ (2 ^ n) + 2 ^ (2 ^ (n-1)) + 1, B (n) = 2 ^ (2 ^ n) - 2 ^ (2 ^ (n-1)) + 1,
Then a (n) = 2 ^ (2 ^ n) + 2 ^ (2 ^ (n-1)) + 1
= 2^(2^n) + 2 * 2^(2^(n-1)) + 1 - 2^(2^(n-1))
= (2^(2^(n-1)) + 1)^2 - (2^(2^(n-2)))^2
= (2^(2^(n-1)) + 1 + 2^(2^(n-2)))*(2^(2^(n-1)) + 1 - 2^(2^(n-2)))
= a(n - 1) * b(n - 1)．
So a (n) = a (n - 1) * B (n - 1) = a (n - 2) * B (n - 2) * B (n - 1)
= ...= a(1) * b(1) * b(2) * ...* b(n -1)．
Obviously, a (n) > 1, B (1),..., B (n - 1) > 1, so a (1), B (1),..., B (n - 1) all have prime factors
Because a (n) - B (n) = 2 * 2 ^ (2 ^ (n-1)),
That is, a (1) * B (1) * B (2) *... * B (n - 1) - B (n) = 2 * 2 ^ (2 ^ (n - 1))
And a (1), B (1),..., B (n - 1), B (n) are all odd numbers,
So the product a (1) B (1)... B (n - 1) and B (n) are coprime
Therefore, each of a (1), B (1),..., B (n - 1) is coprime with B (n)
This shows that for any two terms B (k) and B (J) in {B (n)}, B (k) and B (J) have no common prime factor
Moreover, every term B (k) and a (1) in {B (n)} has no common prime factor
So any two prime factors in a (1), B (1),..., B (n - 1) are different
Therefore, their product a (n) = a (1) * B (1) * B (2) *... * B (n - 1) contains at least n different prime factors
Under the current Gregorian calendar system (if the year is a multiple of 4 but not 100, or if the year is a multiple of 400, then this year is a leap year, with 29 days in February; if the year is not a multiple of 4, or if the year is a multiple of 100 but not 400, then this year is an ordinary year, with 28 days in February). In the next 100 years (2015-2114), new year's day will appear the least on the day of the week. How many times will it appear?
Normally, if this year is a normal year, because the remainder of 365 ／ 7 is 1, the week of the next year will be one more day than this year (this year's Monday, next year's Tuesday, etc.); if there are 366 days in leap year, the next year will need two more days (this year's leap year's Monday, next year's Wednesday, etc.). It is worth noting that 2100 is a normal year, From 2097 to 2104, there are seven times of adding one day in a row. In this way, the new year's day of each year can be calculated from the new year's day of 2015 is Thursday. A better method remains to be studied. The answer is: Monday 14 days, Tuesday 14 days, Wednesday 13 days, Thursday 15 days, Friday 15 days, Saturday 14 days, Sunday 15 days. Obviously, choosing Wednesday 13 times is the least
（1） Xiao Ming plans to finish a book in several days. If he reads 5 pages less every day, he will finish the book 7 / 8 later. If he reads 1 / 3 more pages than he plans, he will finish the book one day earlier?
（2） The original plan was to use several tractors to complete the project within the specified time. If three tractors were added, it would only take 7 / 8 of the specified time to complete the project. If two tractors were reduced, it would be 2 / 3 hours later than the specified time?
Detailed explanation of requirements
It seems that there are some problems in these two questions, because they are not integers. In question 1, let's plan a day, B pages per day, and the total number of pages after reading this book is a * B. according to the meaning of the questions, we can get the equation a * B / (B-5) = a (1 + 7 / 8) a * B / (B + B / 3) = A-1. Solving the equation, we can get a = 4, B = 75 / 7. In question 2, let's plan x tractors, and finish in Y hours
1. Verification: if a figure has only two axes of symmetry, they are perpendicular to each other
2. Let m and n be positive integers, m and n be odd numbers, and (2 ^ (n-1), m) = 1 +m^n)
It is proved that: 1. Let the two symmetrical axes be L1 and L2, and let the symmetrical line of L1 with respect to L2 be L3, then L3 is also the symmetrical axis of the graph. This is because, for the two symmetrical points of L2, the two symmetrical points of L1 must be symmetrical with respect to L3. Therefore, there are only two symmetrical axes known in the title. L3 coincides with L2, and L1 is perpendicular to L2
As shown in the figure, in rectangular ABCD, AE: ed = AF: ab = BG: GC. It is known that the area of △ EFC is 20 and the area of △ FGD is 16. What is the area of rectangular ABCD?
Let AB = CD = a, ad = BC = B on the opposite side of rectangular ABCD, and then let the ratio constant AE: ed = AF: ab = BG: GC = K in the question. If this expression is transformed into the expression of K and the side length a and B of rectangular ABCD, then AE = BG = Kb: (K + 1) ed = GC = BK + 1af = Ka, FB = (1-k) as (rectangular ABCD) = AB = s (RT △ AFE) + s (△ FEC) + s（ RT △ EDC) + s (RT △ FBC) = 12 × AF × AE + 20 + 12 × ed × CD + 12 × FB × BC = 12 × Ka × Kb: (K + 1) + 20 + 12 × B: (K + 1) × a + 12 × (1-k) a × B = 1K + 1 × AB + 20, ab = 20 × (K + 1) K & nbsp; & nbsp; (1) Similarly, s (rectangular ABCD) = AB = s (RT △ FBG) + s (△ FGD) + s (RT △ GDC) + s (RT △ AFD) = 12 × FB × BG + 16 + 12 × GC × CD + 12 × AF × ad = 12 × (1-k) a × K + 1KB + 16 + 12 + BBK + 1 × a + 12 × Ka × B = 2K + 12K + 2 × AB + 16, ab = 32 (K + 1) (2) according to (1) (2), k = 58, substituting (1) or (2), s (rectangular ABCD) = AB = 52cm
As shown in the figure, it is known that e, F, G and H are respectively the midpoint of each side of the square ABCD, and the area of the square ABCD is 80 square centimeters. Calculate the area of the shadow part
80 × 15 = 16 (square centimeter). A: the area of shadow is 16 square centimeter
Draw an inscribed equilateral triangle in the circle, draw another inscribed circle in the equilateral triangle, and draw a second inscribed equilateral triangle in the second circle, so as to continue to draw (as shown in the figure). If the area of the first triangle is 512 square centimeters, then what is the area of the fifth triangle?
According to the stem analysis, the area of the triangle is reduced according to the ratio of 4:1, so 512 △ 4 △ 4 △ 4 = 2 (square centimeter). A: the area of the fifth triangle is 2 square centimeter
The edge length is 4cm, and the surface is all red. Three equal distance cuts are made on each side of the cube, and 64 small cubes are obtained, and the cut planes are all white. Among the 64 small cubes, how many are red on one side, two sides and three sides?
Each side is not red. How many are there?
There are only eight corners on three sides, of course, eight
There are two sides, two on one side. If there are 12 sides, there are 24
One side is in the middle, one side has four, there are 24
There are eight white ones
Three dimensional graphic exercises
The side area of a cylindrical water tower is 157 square meters, the height is 5 cm, the diameter of the bottom surface is how many meters? How many square meters does it cover? Please list the formula,
The water tower is 5 meters high!
Bottom perimeter C = 157 / 5m = 31.4m
Bottom diameter d = 31.4 / 3.14m = 10m
Occupied area s = 3.14x5x5 M2 = 78.5 M2
In the shopping malls A and B, the students found that the unit price of the sportswear he liked was the same, and the unit price of the sports shoes was the same. The sum of the unit price of the sports shoes and the sports shoes was 452 yuan, and the unit price of the sports shoes was 8 yuan less than 4 times of the unit price of the sports shoes
[1] How much is the unit price of sports shoes and sneakers
[2] One day, the student went to the street just in time to catch up with the store promotion. All the goods of a were sold at a 20% discount, and every 100 yuan of B was returned with 30 yuan coupon. He only brought 400 yuan. If he only bought these two items in one shopping mall, can you explain which shopping mall he can buy them in? If two shopping malls can, which one will save more money
1. The unit price of shoes is x yuan, and that of sportswear is 4x-8 yuan
4x-8+x=452
5x=460
x=92
Shoes 92 yuan
Sportswear 360 yuan
A: 452 × 0.8 = 361.6
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