# The number 4,000,000 has 63 positive integral factors. How do you find a and b, where 2^a 5^b is the product of all positive factors of 4,000,000?

By signing up, you agree to our Terms of Service and Privacy Policy

Take note of this:

Below, we make use of these...

The sum of the individual factors is:

By signing up, you agree to our Terms of Service and Privacy Policy

To find ( a ) and ( b ), where ( 2^a \times 5^b ) is the product of all positive factors of ( 4,000,000 ), first, you need to express ( 4,000,000 ) in terms of its prime factors.

( 4,000,000 = 2^6 \times 5^6 )

Next, to determine the exponents ( a ) and ( b ), observe that each factor can either include a ( 2 ) or a ( 5 ), or both. Since ( 4,000,000 ) has ( 63 ) positive integral factors, and it can be represented as ( 2^6 \times 5^6 ), each factor could have any combination of the prime factors ( 2 ) and ( 5 ) with exponents ranging from ( 0 ) to ( 6 ).

Given that ( 63 ) is the total number of factors, there are ( 7 ) choices for the exponent of ( 2 ) (( 0 ) through ( 6 )) and ( 7 ) choices for the exponent of ( 5 ) (( 0 ) through ( 6 )).

Therefore, the total number of combinations of ( a ) and ( b ) is ( 7 \times 7 = 49 ).

As ( 2^6 \times 5^6 ) is included in this count, subtract ( 1 ) to get the actual number of combinations of ( a ) and ( b ).

Thus, ( a ) can be any integer from ( 0 ) to ( 6 ), and ( b ) can also be any integer from ( 0 ) to ( 6 ).

Therefore, ( a ) can take on ( 7 ) different values, and ( b ) can also take on ( 7 ) different values, giving a total of ( 7 \times 7 = 49 ) combinations.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you simplify the square root #sqrt49#?
- How do you evaluate #(4\times 10^ { - 6} ) + ( 0.08\times 10^ { - 2} )#?
- What is 3 to the 8th power?
- How do you combine #\frac { w } { 5} root3(-64) + root3{ 512w ^ { 3 } } / { 5} - \frac { 2} { 5} \sqrt { 50w } - 4\sqrt { 2w }# into a single term, if possible?
- How do you evaluate #\frac { 10+ 2( - 5) ^ { 2} } { ( 2^ { 2} ) ( 3) }#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7