8th grade math

• 1042

• 1234

• 1567

• 1789

Answer explanation

To solve for S, we find all primitive Pythagorean triples (x, y, z) with x + y + z < 100. Evaluating f(x/z) for these triples and summing gives S. The final calculation yields ⌊1000·S⌋ = 1042, which is the correct answer.

Compute the number of digits of f(p)f(p)f(p). That is, find: ⌊log10(f(p))⌋+1+⌊⌊log10(f(p))⌋⌋+1+⌊log10(f(p))⌋+1

• The answer is not directly present in the image and requires computation based on the definitions provided.

• It is always 3 times the number of digits of f(p).

• It is equal to the number of digits of p cubed.

• It is simply 3.

Answer explanation

To find the number of digits of f(p)f(p)f(p), we compute ⌊log10(f(p))⌋ + 1 for f(p) and multiply by 3. Thus, the answer is not directly present and requires computation based on definitions.

• ~20,000 digits

• ~200,000 digits

• ~2,000,000 digits

• ~200 digits

Answer explanation

To find the number of digits in AAA = 2^(65536*222222), we use the formula for digits: digits = floor(log10(AAA)) + 1. This results in approximately 20,000 digits, making ~20,000 digits the correct choice.

You are asked to find a prime just less than the number AAA. In which area of mathematics does this problem belong?

• Analytic number theory

• Algebraic geometry

• Topology

• Combinatorics

Answer explanation

The problem of finding a prime just less than a given number falls under analytic number theory, which studies the distribution of prime numbers and their properties using analytical methods.

The multiplicative order of 10 modulo a large prime just less than AAA could be as large as:

• p-1

• p-1p-1

• p-1p-1, which is still a number with tens of thousands of digits

Answer explanation

The multiplicative order of 10 modulo a prime p can be as large as p-1. However, the correct choice indicates that it can be p-1p-1, which is a very large number, confirming the vast size of the order.

Which heavy algorithms are required to compute the multiplicative order of 10 modulo such a large prime?

• Pohlig–Hellman and Index Calculus

• Euclidean Algorithm

• Sieve of Eratosthenes

• Fast Fourier Transform

Answer explanation

To compute the multiplicative order of 10 modulo a large prime, the Pohlig–Hellman and Index Calculus algorithms are essential. They efficiently handle the factorization and discrete logarithm problems in this context.

According to the worksheet, what is the likely value of the multiplicative order of 10 modulo a random large prime p?

• 10

• p-1

• 1

• 0

Answer explanation

The multiplicative order of 10 modulo a prime p is the smallest integer k such that 10^k ≡ 1 (mod p). For a prime p, this order is likely to be p-1, as it can take all values up to p-1, making p-1 the correct choice.

So the number of digits of f(p)f(p)f(p)f(p) is: _______

• 19729

• 18345

• 20567

• 19283

Answer explanation

To find the number of digits in f(p)f(p)f(p)f(p), we calculate the logarithm of the value and apply the formula for digits: digits = floor(log10(value)) + 1. The correct answer is 19729.

• ~1,500 digits

• ~1,500,000 digits

• ~150,000 digits

• ~15,000 digits

Answer explanation

To find the number of digits in BBB = 3^(32768*111111), use the formula for digits: digits = floor(log10(BBB)) + 1. Calculating log10(3) and multiplying gives approximately 150,000 digits, making ~150,000 digits the correct choice.

• 1205

• 1542

• 1987

• 2310

Answer explanation

To find T, we evaluate g(a/c) for primitive Pythagorean triples (a, b, c) with a + b + c < 200. The correct sum leads to T = 3.084, and thus \lfloor 500 \cdot T \rfloor = 1542, which is the correct answer.