Which are prime numbers, which are composite numbers, which are even numbers and which are odd numbers in 49,51,12,37,1,0.5,85

Which are prime numbers, which are composite numbers, which are even numbers and which are odd numbers in 49,51,12,37,1,0.5,85

The prime number is 51.37
The total is 49.12.85
Even 12
Odd 49.51.37.1.85

Find less than 100 only 15 divisors of the largest natural number, the process said

X has no solution within 100
Lemma:
If x = a ^ m * B ^ n (a, B are prime numbers)
Take any divisor y, y = a ^ p * B ^ Q (P is 0-m integer, q is 0-n integer)
Then x has (M + 1) * (n + 1) divisors
Let the number be x, 15 = 1 * 15 = 3 * 5
From the lemma, there are two possibilities
x=a^14 or
x=a^2*b^4
Let a = 3, B = 2, then x = 144
Let a = 2, B = 3, then x = 324 > 100
So the minimum is 144

What is the minimum natural number with the sum of all positive divisors equal to 15?

Divisor sum formula: let a = P1 ^ M1 * P2 ^ M2 *... PN ^ Mn
Sum of divisors = (1 + P1 ^ 1 + P1 ^ 2... + P1 ^ M1) * (1 + P2 +... + P2 ^ m2) *... * (1 + PN +... + PN ^ Mn)
Note that 15 = 3 * 5,3 = 1 + 2 ^ 1, but 5 cannot be expressed as the sum of 1 and the power of 1 of a prime number
Only 15 = 1 + 2 + 2 ^ 2 + 2 ^ 3
So the natural number is 2 ^ 3 = 8

The sum of all divisors of a natural number
What is the sum of all constraints of 240?
240 = 4 power of 2 × 1 power of 3 × 1 power of 5
For the sum of divisors, use (0 power of 2 + 1 power of 2 + 2 power of 2 + 3 power of 2 + 4 power of 2 + 5 power of 2) × (0 power of 3 + 1 power of 3) × (0 power of 5 + 1 power of 5) = 744
I only know this formula, but why do I do it? Why on earth?
Thank you for your understanding!

It's much more comfortable to understand with smaller numbers. For example, if you want to find the sum of all the divisors of 6, you can know that it's 1 + 2 + 3 + 6 by oral arithmetic
Use your formula to calculate
6 = 1 power of 2 × 1 power of 3
(0 power of 2 + 1 power of 2) × (0 power of 3 + 1 power of 3)
=（1+2）×（1+3）
=1×1+1×3+2×1+2×3
=1+3+2+6
You can see that your formula, if expanded, is actually the sum of 1, 2, 3 and 6
Of course, 240 is the same, but it is bigger after expansion, but it must be the sum of all divisors. Do you understand?

What is the minimum natural number with only 15 divisors?

15=(2+1)×(4+1)=14+1
So the minimum is 2 ^ 4 × 3, or 2 ^ 14
So the minimum is 2 ^ 4 × 3 & # 178; = 144

(1) What is the minimum natural number that the sum of all positive divisors equals 15? (2) what is the minimum natural number that the product of all positive divisors equals 64?

8: The positive divisor is 1,2,4,8; 1 + 2 + 4 + 8 = 16
8: The positive divisor is 1,2,4,8; 1 * 2 * 4 * 8 = 64

The minimum divisor of any natural number is 1
The minimum divisor of any natural number is 1. Now 0 is a natural number
Who said √ or ×?

Yes
It is meaningless to divide by zero

The natural number a has 10 divisors, including 1 and itself. What is the product of these divisors?

There are innumerable such numbers, if the smallest, 2 * 2 * 2 * 3
Sum of all factors = (1 + 2 + 4 + 8 + 16) * (1 + 3) = 124

Given two natural numbers, their least common multiple is 36 and the greatest common divisor is 24. Find out the product of the two numbers. Find out what are the two numbers?

These two numbers do not exist

Write the greatest common factor and least common multiple of the following groups directly. 8 and 64

The greatest common factor is 8 and the least common multiple is 64