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 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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Answer 1 · 2016-06-21T06:36:50

$a = 8$ $a=8$ and $b = 6$ $b=6$

Explanation:

$4000000 = 4 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10$ $4000000=4xx10xx10xx10xx10xx10xx10$

= $2^{2} \times 10^{6}$ $2^2xx10^6$

= $2^{2} \times {(2 \times 5)}^{6}$ $2^2xx(2xx5)^6$

= $2^{2} \times 2^{6} \times 5^{6}$ $2^2xx2^6xx5^6$

= $2^{2 + 6} \times 5^{6}$ $2^(2+6)xx5^6$

= $2^{8} \times 5^{6}$ $2^8xx5^6$

Comparing it with $2^{a} 5^{b}$ $2^a5^b$ , we get

$a = 8$ $a=8$ and $b = 6$ $b=6$

Answer 2 · 2016-06-21T15:52:09

$a = 252$ $a=252$ and $b = 189$ $b=189$

Explanation:

$4000000 = 4 \cdot 10^{6} = 2^{8} \cdot 5^{6}$ $4000000 = 4 * 10^6 = 2^8*5^6$

So each of the positive integral factors of $4000000$ $4000000$ is of the form $2^{m} \cdot 5^{n}$ $2^m*5^n$ where $m \in {0, 1, 2, 3, 4, 5, 6, 7, 8}$ $m in { 0, 1, 2, 3, 4, 5, 6, 7, 8 }$ and $n \in {0, 1, 2, 3, 4, 5, 6}$ $n in {0, 1, 2, 3, 4, 5, 6}$ .

Note that:

$8 \sum m = 0 m = \frac{1}{2} 8 (8 + 1) = 36$ $sum_(m=0)^8 m = 1/2 8 (8+1) = 36$

$6 \sum n = 0 n = \frac{1}{2} 6 (6 + 1) = 21$ $sum_(n=0)^6 n = 1/2 6 (6+1) = 21$

We use these below...

The product of all the factors is:

$8 \prod m = 0 (6 \prod n = 0 2^{m} \cdot 5^{n})$ $prod_(m=0)^8 (prod_(n=0)^6 2^m*5^n)$

$= 8 \prod m = 0 (2^{7 m} 6 \prod n = 0 5^{n})$ $= prod_(m=0)^8 (2^(7m) prod_(n=0)^6 5^n)$

$= (8 \prod m = 0 2^{7 m}) {(6 \prod n = 0 5^{n})}^{9}$ $= (prod_(m=0)^8 2^(7m)) (prod_(n=0)^6 5^n)^9$

$= 2^{7 \sum_{m = 0}^{8} m} \cdot 5^{9 \sum_{n = 0}^{6} n}$ $= 2^(7sum_(m=0)^8 m)*5^(9 sum_(n=0)^6 n$

$= 2^{7 \cdot 36} \cdot 5^{9 \cdot 21}$ $= 2^(7*36)*5^(9*21)$

$= 2^{252} \cdot 5^{189}$ $= 2^252 * 5^189$

So $a = 252$ $a=252$ and $b = 189$ $b=189$

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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?

2 Answers

Explanation:

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