How many positive factors does
62x(63^3+63^2+63+1)+1 have?
(With explanatio pls... Tnx)
63^3+63^2+63+1 = (63^4-1)/(63-1) = 63^4-1
So, your number is
63^4-1+1 = 63^4 = 3^8 * 7^4
So, the prime factors are 3 and 7
I'll leave it to you to determine how many factors there are in all
Tnx... But where did exponent 4 and minus one come from?
To find the number of positive factors of the given expression, we can factorize it and then count the factors.
Let's break down the expression inside the parentheses:
63^3 + 63^2 + 63 + 1
To simplify, we start by factoring out 63 from each term:
63^3 + 63^2 + 63 + 1 becomes 63(63^2 + 63 + 1) + 1
Now, we can see that 63 is a factor common to each term inside the parentheses. We can factor it out:
63(63^2 + 63 + 1) + 1 becomes 63(1 + 63 + 63^2) + 1
Now, let's simplify the expression:
63(1 + 63 + 63^2) + 1
= 63(1 + 63 + 3969) + 1
= 63(4033) + 1
= 254079 + 1
= 254080
Now, we have the simplified expression: 62x(254080) + 1.
To find the number of positive factors, we need to factorize the expression further and then count the factors.
The expression 62x(254080) + 1 can be factored as:
62(2^1)x(5^1)x(47^1)x(271^1) + 1
The exponents represent the power to which each prime factor is raised.
To calculate the number of factors, we add 1 to each exponent and multiply them together.
For this expression:
Power of 2 = 1 + 1 = 2
Power of 5 = 1 + 1 = 2
Power of 47 = 1 + 1 = 2
Power of 271 = 1 + 1 = 2
Now, multiply the exponents together:
2 x 2 x 2 x 2 = 16
So, the given expression 62x(63^3+63^2+63+1)+1 has 16 positive factors.