1、1、2、3、5、8、13…… . 90 numbers in a row, then, what is the remainder of the sum of these 90 numbers divided by 5

1、1、2、3、5、8、13…… . 90 numbers in a row, then, what is the remainder of the sum of these 90 numbers divided by 5

This is the famous Fibonacci sequence
A(n+2)=A(n+1)+A(N)
The sum of any four terms is as follows:
A(n+4) +A(n+3)+A(n+2)+A(n+1)
=A(n+3)+A(n+2)+A(n+3)+A(n+2)+A(n+1)
=2*A(n+3) +2*A(n+2)+A(n+1)
=2*A(n+2)+2*A(n+1) +2*A(n+2)+A(n+1)
=4A(n+2)+3A(n+1)
=4*[A(n+1)+A(n)]+3A(n+1)
=7A(n+1)+4A(n)
=11A(n)+7A(n-1)
=[10A(n)+5A(n-1)]+A(n-1)+[A(n)+A(n-1)]
=[10A(n)+5A(n-1)]+A(n-2)+A(n-3)+[A(n)+A(n-1)]
=5*[2*A(n)+A(n-1)]+A(n-2)+A(n-3)+A(n)+A(n-1)
The sum of 6 terms = 2 + 3 + 5 + 8 = 18, and the remainder is 3
The sum of 10 terms and 3 ~ 6 terms is related to 5 congruence: the remainder is 3
Similarly:
Divide the sum of 11 ~ 14 items by 5 to get 3
.
Divide the sum of 87 ~ 90 items by 5 to get 3
So the sum of 3 ~ 90 terms and (3 * [(90-2) / 4] = 66) are congruent with 5, that is, the remainder is 1
The remainder of 5 is (1 + 1 + 1) = 3
Result: the sum of these 90 numbers divided by 5 is 3

What's the power of 8 of 0.125 multiplied by the power of 99 of 8

This seems to be the number one. Let me try to answer it
0.125^8*8^99
=0.125^8*8^8*8^91
=(0.125*8)^8*8^91
=1^8*8^91
=8^91
=2^273
At least the order of magnitude is above the 80th power of 10

How many remainders of 3 to 10000?
The answer on the first floor is not correct. It is that all n are infinite. For the remainder of 10000, it is not less than 10000

3,9,27,81,243,729,2187,6561,9683,9049,7147,1441,4323,2969,8907,6721,163,489,1467,4401,3203,9609,8827,6481,9443,8329,4987,4961,4883,4649,3947,1841,5523,6569,9707,9121,7363,2089,6267,8801,6403,9209,7627,2881,8643,5929,7787,3361,83,249,747,2241,6723,169,507,1521,4563,3689,1067,3201,9603,8809,6427,9281,7843,3529,587,1761,5283,5849,7547,2641,7923, 3769,1307,3921,1763,5289,5867,7601,2803,8409,5227,5681,7043,1129,3387,161,483,1449,4347,3041,9123,7369,2107,6321,8963,6889,667,2001,6003,8009,4027,2081,6243,8729,6187,8561,5683,7049,1147,3441,323,969,2907,8721,6163,8489,5467,6401,9203,7609,2827,8481,5443,6329,8987,6961,883,2649,7947,3841,1523,4569,3707,1121,3363,89,267,801,2403,7209,1627,4881, 4643,3929,1787,5361,6083,8249,4747,4241,2723,8169,4507,3521,563,1689,5067,5201,5603,6809,427,1281,3843,1529,4587,3761,1283,3849,1547,4641,3923,1769,5307,5921,7763,3289,9867,9601,8803,6409,9227,7681,3043,9129,7387,2161,6483,9449,8347,5041,5123,5369,6107,8321,4963,4889,4667,4001,2003,6009,8027,4081,2243,6729,187,561,1683,5049,5147,5441,6323,8969, 6907,721,2163,6489,9467,8401,5203,5609,6827,481,1443,4329,2987,8961,6883,649,1947,5841,7523,2569,7707,3121,9363,8089,4267,2801,8403,5209,5627,6881,643,1929,5787,7361,2083,6249,8747,6241,8723,6169,8507,5521,6563,9689,9067,7201,1603,4809,4427,3281,9843,9529,8587,5761,7283,1849,5547,6641,9923,9769,9307,7921,3763,1289,3867,1601,4803,4409,3227,9681, 9043,7129,1387,4161,2483,7449,2347,7041,1123,3369,107,321,963,2889,8667,6001,8003,4009,2027,6081,8243,4729,4187,2561,7683,3049,9147,7441,2323,6969,907,2721,8163,4489,3467,401,1203,3609,827,2481,7443,2329,6987,961,2883,8649,5947,7841,3523,569,1707,5121,5363,6089,8267,4801,4403,3209,9627,8881,6643,9929,9787,9361,8083,4249,2747,8241,4723,4169,2507, 7521,2563,7689,3067,9201,7603,2809,8427,5281,5843,7529,2587,7761,3283,9849,9547,8641,5923,7769,3307,9921,9763,9289,7867,3601,803,2409,7227,1681,5043,5129,5387,6161,8483,5449,6347,9041,7123,1369,4107,2321,6963,889,2667,8001,4003,2009,6027,8081,4243,2729,8187,4561,3683,1049,3147,9441,8323,4969,4907,4721,4163,2489,7467,2401,7203,1609,4827,4481,3443, 329,987,2961,8883,6649,9947,9841,9523,8569,5707,7121,1363,4089,2267,6801,403,1209,3627,881,2643,7929,3787,1361,4083,2249,6747,241,723,2169,6507,9521,8563,5689,7067,1201,3603,809,2427,7281,1843,5529,6587,9761,9283,7849,3547,641,1923,5769,7307,1921,5763,7289,1867,5601,6803,409,1227,3681,1043,3129,9387,8161,4483,3449,347,1041,3123,9369,8107,4321, 2963,8889,6667,1
Use Excel: C14 = if (3 * C13 > 10000, mod (3 * c1310000), 3 * C13), and then drag to get the 500 non repeating numbers

A classic number theory problem,
If ax (0) + by (0) is the smallest integer of the form ax + by (x, y are any integers, a, B are two integers which are not all zero), then [ax (0) + by (0)] | (AX + by)
Give a more detailed proof, thank you

In this paper, we prove that ax + by is an integer, and ax0 + by0 > 0. By division with remainder, ax + by = (ax0 + by0) Q + R (q is an integer, 0 ≤ r < ax0 + by0).. r = ax + by - (ax0 + by0) q = a (x-x0q) + B (y-y0q), we show that R is also a number in the form of AX + by, but ax0 + by0 is a set of numbers in the form of AX + by

ACM number theory topic a ^ B ^ C mod 100000007 how to fast power. Data range of three numbers are less than 1000000000

#includetypedef __ int64 lld;lld mod(lld a,lld b,lld m){\x05lld ret=1;\x05a%=m;\x05while(b)\x05{\x05\x05if(b&1)ret=ret*a%m;\x05\x05b>>=1;\x05\x05a=a*a%m;\x05\x05//printf("b=%I64d\n",b);\x05}\x05return ...

Who can solve the problem of number theory
1 / a = 1 / x + 1 / y + 1 / Z, a is a constant, and a, x, y, Z are all positive integers. Find all solutions of X, y, Z
After thinking for a long time, I couldn't solve it
If you think this problem can't be solved, please explain the reason
This is a question I want to answer,

It can't be said that it can't be solved, but the answer can't be expressed normally at all
For example, consider the equation M / N = 1 / x + 1 / y, that is, MXY = n (x + y)
We can get (mx-n) (my-n) = n ^ 2
In order to obtain all its solutions, we need to consider the method of decomposing n ^ 2 into the product of two positive integers
For example, when n = 6, 36 = 1 * 36 = 2 * 18 = 3 * 12 = 4 * 9 = 6 * 6 = 9 * 4 = 12 * 3 = 18 * 2 = 36 * 1
Generally, every n ^ 2 = PQ can get x = (n + P) / m, y = (n + Q) / m. when x and y are integers, it is a solution,
All solutions can be obtained in this way
The original problem 1 / a = 1 / x + 1 / y + 1 / Z will be a lot of trouble, the general method might as well assume X

What is the meaning of power?

The result of power is called power
Represents the product of several identical factors

What is the meaning of power?
If power is a multiplication operation, what is power and power?

What you want is the following content. I hope it can be useful for you to calculate power. 1. Teaching content: 1. Multiplication of the same base power. 2. Power and product power. 3. Division of the same base power. 2. Skill requirements: master the operation properties of positive integer power (multiplication of the same base power, power, product power, division of the same base power)

Do the third power of (- 2) and - 2 have the same meaning? What are their respective meanings?
What is the meaning of them?

The third power of (- 2) is the power operation of - 2, while the third power of - 2 is the power operation of 2 first, and then take the opposite number as the result. The two results are the same, but the meaning is different
Hope to adopt

What does the - 7th power of 10 mean

One tenth of the seventh power