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?
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The sum of the individual factors is:
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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.
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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.
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.
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