4^17 - 2^28
= 2^34 - 2^28
= 2^28 (2^6 - 1)
= 2^28 (63)
= 2^28 * 9 * 7
Prime factors are 2, 3 and 7. Greatest prime factor, hence, is 7
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Greatest prime factor...
Source: Beat The GMAT — Problem Solving |
-
jayhawk2001
- Community Manager
- Posts: 789
- Location: Silicon valley, California
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Mathematically, Prime factors originated from prime number. prime numbers which can be defined as any number that is divisible by itself and 1 alone. examples are 2, 3, 5, 7, 11 etc.
To every number, there are always factors which constitute the number. prime factors is finding which prime number we need to multiply to get the original numbers.
for the question, 4^17 - 2^28.
2^(2*17) - 2^28
2^34 - 2^28
2^28 (2^6 - 1)
2^28 * (64-1)
2^28 * (63)
2^28 * (7*9)
2^28 * (7 * 3^2)
2^28 * 3^2 * 7
for this solution, the prime factors are 2, 3 and 7. the greatest prime factor is the highest which of course is 7.
To every number, there are always factors which constitute the number. prime factors is finding which prime number we need to multiply to get the original numbers.
for the question, 4^17 - 2^28.
2^(2*17) - 2^28
2^34 - 2^28
2^28 (2^6 - 1)
2^28 * (64-1)
2^28 * (63)
2^28 * (7*9)
2^28 * (7 * 3^2)
2^28 * 3^2 * 7
for this solution, the prime factors are 2, 3 and 7. the greatest prime factor is the highest which of course is 7.
To find the greatest prime factor of a number, we want to do a prime factorization - in other words, we want to break the number down as small as it'll go. But to do that, we first have to combine our terms.
When we see 4^17 - 2^28, we should notice right away that we can simplify 4 to 2^2. This will give us the same base for both of our terms (in this case, 2), so let's go ahead and do that. We'll need to remember our power rule: (x^m)^n = x^mn.
(2^2)^17 - 2^28
2^34 - 2^28
Now, we know that we can't add or subtract terms with the same base unless they *also* have the same exponent, so we'll have to find another way to simplify them. Thinking about the product rule ((x^m)(x^n) = x^(m+n)), we should recognize that 2^34 = 2^(28+6) = (2^28)(2^6). This means that we can pull 2^28 out of both terms:
(2^28)(2^6) - 2^28
(2^28)(2^6 - 1)
Now 2^28 has been factored as much as possible. We also have smaller numbers to subtract (2^6 and 1), so we can go ahead and combine them:
(2^28)(64 - 1)
(2^28)(63)
Then finally, we can do prime factorization on 63:
(2^28)(7)(9)
(2^28)(7)(3^2)
So the prime factors are 2, 3, and 7. Of these, the greatest prime factor is 7.
When we see 4^17 - 2^28, we should notice right away that we can simplify 4 to 2^2. This will give us the same base for both of our terms (in this case, 2), so let's go ahead and do that. We'll need to remember our power rule: (x^m)^n = x^mn.
(2^2)^17 - 2^28
2^34 - 2^28
Now, we know that we can't add or subtract terms with the same base unless they *also* have the same exponent, so we'll have to find another way to simplify them. Thinking about the product rule ((x^m)(x^n) = x^(m+n)), we should recognize that 2^34 = 2^(28+6) = (2^28)(2^6). This means that we can pull 2^28 out of both terms:
(2^28)(2^6) - 2^28
(2^28)(2^6 - 1)
Now 2^28 has been factored as much as possible. We also have smaller numbers to subtract (2^6 and 1), so we can go ahead and combine them:
(2^28)(64 - 1)
(2^28)(63)
Then finally, we can do prime factorization on 63:
(2^28)(7)(9)
(2^28)(7)(3^2)
So the prime factors are 2, 3, and 7. Of these, the greatest prime factor is 7.
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More legibly 
4¹� - 2²� =>
(2²)¹� - 2²� =>
2³� - 2²� =>
2²�*2� - 2²�*1 =>
2²� * (2� - 1) =>
2²� * 63 =>
2²� * 3 * 3 * 7
4¹� - 2²� =>
(2²)¹� - 2²� =>
2³� - 2²� =>
2²�*2� - 2²�*1 =>
2²� * (2� - 1) =>
2²� * 63 =>
2²� * 3 * 3 * 7
GMAT/MBA Expert
-
Scott@TargetTestPrep
- GMAT Instructor
- Posts: 8138
- Location: Los Angeles, CA
- Thanked: 43 times
We need to determine the greatest prime factor of 4^17 – 2^28. We can start by breaking 4^17 into prime factors.
4^17 = (2^2)^17 = 2^34
Now our equation is as follows:
2^34 – 2^28
Note that the common factor in each term is 2^28; thus, the expression can be simplified as follows:
2^28(2^6 – 1)
2^28(64 – 1)
2^28(63)
2^28 x 9 x 7
2^28 x 3^2 x 7
We see that the greatest prime factor must be 7.
Scott Woodbury-Stewart
Founder and CEO
[email protected]
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