Prove that, 36^7−6^13 is divisible by 30.
To prove that 36^7 - 6^13 is divisible by 30, we need to show that the difference is divisible by both 2 and 3, since 30 is divisible by both 2 and 3.
Step 1: Express the numbers as factors of 2 and 3.
36 can be written as 2^2 * 3^2 and 6 can be written as 2 * 3.
Step 2: Rewrite the expression using the factorization.
36^7 - 6^13 can be written as (2^2 * 3^2)^7 - (2 * 3)^13.
Step 3: Apply the exponent rule for multiplication.
(2^2 * 3^2)^7 - (2 * 3)^13 can be rewritten as (2^14 * 3^14) - (2^13 * 3^13).
Step 4: Apply the distributive property.
(2^14 * 3^14) - (2^13 * 3^13) can be rewritten as 2^14 * 3^14 - 2^13 * 3^13.
Step 5: Factor out the common factors.
2^14 * 3^14 - 2^13 * 3^13 can be rewritten as 2^13 * 3^13 * (2 * 3 - 1).
Step 6: Simplify.
The expression 2^13 * 3^13 * (2 * 3 - 1) is divisible by both 2 and 3, because it contains factors of both 2 and 3. Therefore, 36^7 - 6^13 is divisible by 30.
To prove that 36^7 - 6^13 is divisible by 30, we need to show that it is divisible by both 2 and 3, since 30 is divisible by both 2 and 3.
First, let's consider the divisibility by 2. Since 36 is divisible by 2, we can write 36 as 2 * 18. Similarly, 6 can be written as 2 * 3. So the equation becomes:
(2 * 18)^7 - (2 * 3)^13
Using the properties of exponents, we can simplify this expression. When we raise a product to a power, we can distribute the power to each factor. So we have:
2^7 * 18^7 - 2^13 * 3^13
Now, let's look at the first term, 2^7 * 18^7. Since 2^7 contains a factor of 2 to the power of 1 (2^1), it is divisible by 2. Also, 18^7 is divisible by 2 because 18 is divisible by 2.
Next, let's evaluate the second term, 2^13 * 3^13. Both terms contain a factor of 2^13, so this term is clearly divisible by 2.
So far, we have shown that both terms in the expression, (2 * 18)^7 - (2 * 3)^13, are divisible by 2.
Now let's consider the divisibility by 3. We can factor 36 as 3 * 12, and 6 as 3 * 2. So the equation becomes:
(3 * 12)^7 - (3 * 2)^13
Using the same process as before, we can simplify this expression:
3^7 * 12^7 - 3^13 * 2^13
Looking at the first term, 3^7 * 12^7, we see that 3^7 is divisible by 3, and 12^7 is divisible by 3 since 12 is divisible by 3.
For the second term, 3^13 * 2^13, both terms contain a factor of 3^13, so the second term is divisible by 3.
Therefore, we have shown that both terms in the expression, (3 * 12)^7 - (3 * 2)^13, are divisible by 3.
Since the expression is divisible by both 2 and 3, it must be divisible by their least common multiple (LCM), which is 6.
Hence, we have proved that 36^7 - 6^13 is divisible by 6.
take out the HCF
36^7−6^13
= 6^14 - 6^13
= 6^13(6-1)
= 5(6^13)
clearly 6^13 is divisible by 6
and 5(6^13) is divisible by 5, so
5(6^13) is divisible by 5*6 or 30