How many positive factors does
62x(63^3+63^2+63+1)+1 have?
(with solution pls... thanks...)
63^3 + 63^2 + 63 + 1
= 63^2(63 + 1) + (63+1)
= (63^2 + 1)(63+1)
= 3970 x 64
62x(63^3+63^2+63+1)+1
= 62 x 3970 x 64 + 1
= 15752961
= 2401x6561
= 2401 x 3^8
= 7x343x3^8
= 7^4 x 3^8
So now we need to take different combinations of factors, e.g. 7x7 x 3x3x3 would be a factor
the four factors of 7 can be taken in 5 ways,
that is, take none, take one, take 2, take 3 or take 4 of them
in the same way, the eight 3's can be taken in 9 ways.
So the total number of subsets of the above is
5x9 or 45
BUT, that would include taking neither the 7 nor the 3
so we subtract 1
number of positive factors is 44
To find the number of positive factors that the expression
62x(63^3+63^2+63+1)+1 has, we need to simplify the expression first.
Let's start by simplifying the expression inside the parentheses, 63^3+63^2+63+1:
63^3 = 63 * 63 * 63
63^2 = 63 * 63
The expression now becomes:
62x(63 * 63 * 63 + 63 * 63 + 63 + 1) + 1
Next, we can simplify the expression inside the parentheses further:
63 * 63 * 63 = 63^3 = 250,047
63 * 63 = 63^2 = 3969
The expression can now be written as:
62x(250,047 + 3969 + 63 + 1) + 1
Simplifying the addition inside the parentheses:
250,047 + 3969 + 63 + 1 = 254,080
Substituting this back into the expression:
62x(254,080) + 1
Now, we can find the number of positive factors. To do this, we need to factorize 254,080.
First, let's factor out the power of 2:
254,080 = 2^8 * 5 * 397
Now, we can find the number of factors. The formula is:
Number of factors = (p1 + 1) * (p2 + 1) * ... * (pn + 1)
where p1, p2, ..., pn are the exponents of the prime factors.
In this case, we have:
Number of factors = (8 + 1) * (1 + 1) * (1 + 1)
Number of factors = 9 * 2 * 2
Number of factors = 36
Therefore, the expression 62x(63^3+63^2+63+1)+1 has 36 positive factors.