Prove the divisibility of the following numbers:
12^8·9^12 by 6^16
12^8·9^12
= 3^8 * 4^8 * 3^24
= 3^32 * 2^16
= 3^16 * 2^16 * 3^16
= 6^16 * 3^16
which is clearly divisible by 6^16
To prove the divisibility of 12^8·9^12 by 6^16, we need to show that 6^16 is a factor of 12^8·9^12.
Step 1: Express 12 and 9 in terms of their prime factorization:
12 = 2^2 · 3
9 = 3^2
Step 2: Substitute the prime factorization into the expression:
12^8·9^12 = (2^2 · 3)^8 · (3^2)^12
Step 3: Simplify the expression:
(2^2 · 3)^8 · (3^2)^12 = (2^16 · 3^8) · (3^24) = 2^16 · 3^8 · 3^24
Step 4: Multiply the powers of 3:
3^8 · 3^24 = 3^(8+24) = 3^32
Step 5: Substitute back into the expression:
2^16 · 3^8 · 3^24 = 2^16 · 3^32
Since 6 = 2^1 · 3^1, we can express 6^16 as (2^1 · 3^1)^16 = 2^16 · 3^16.
Step 6: Compare the two expressions:
2^16 · 3^32 is divisible by 2^16 · 3^16 if the exponent of 2 and the exponent of 3 in 2^16 · 3^32 are both greater than or equal to the respective exponents in 2^16 · 3^16.
In this case, the exponent of 2 in 2^16 is equal to 16, which is greater than or equal to the exponent of 2 in 2^16 · 3^16, which is 16. Similarly, the exponent of 3 in 3^32 is 32, which is also greater than or equal to the exponent of 3 in 2^16 · 3^16, which is 16.
Therefore, 6^16 is a factor of 12^8·9^12, which proves that 12^8·9^12 is divisible by 6^16.
To prove the divisibility of the numbers 12^8 · 9^12 by 6^16, we need to show that the quotient is a whole number, that is, there is no remainder.
First, let's break down each number to identify their prime factors:
12^8 = (2^2 · 3)^8
9^12 = (3^2)^12
6^16 = (2 · 3)^16
Now, we can rewrite the expression as follows:
(2^2 · 3)^8 · (3^2)^12 ÷ (2 · 3)^16
To prove divisibility, we need to compare the exponents of the common prime factors. Let's do that for each factor.
For the prime factor 2:
The exponent in 12^8 is 2^8 = 256.
The exponent in 6^16 is (2 · 3)^16 = 2^16 = 65536.
256 is less than 65536, meaning that the exponent of 2 in 12^8 · 9^12 is less than the exponent of 2 in 6^16.
For the prime factor 3:
The exponent in 12^8 is (2^2 · 3)^8 = 3^8 = 6561.
The exponent in 9^12 is (3^2)^12 = 3^24 = 6561.
The exponent in 6^16 is (2 · 3)^16 = 3^16 = 43046721.
6561 is less than 43046721, meaning that the exponent of 3 in 12^8 · 9^12 is less than the exponent of 3 in 6^16.
Since the exponents of both prime factors, 2 and 3, are less in 12^8 · 9^12 compared to 6^16, the expression is divisible.