Let’s tackle this problem step by step. The equation given is:

1 + 2 * 3 + 4 * 5 + 6 * 7 + 8 * 9 = 479

But it’s incorrect, and I need to add a pair of parentheses to make it true. Okay, first things first, I should understand what the current value of the left side is without any parentheses.

Let’s recall the order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). So, multiplication comes before addition.

So, calculating step by step:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

Now, plugging these back into the equation:

1 + 6 + 20 + 42 + 72

Adding these up:

1 + 6 = 7

7 + 20 = 27

27 + 42 = 69

69 + 72 = 141

So, without any parentheses, the left side equals 141, but it should be 479. That’s quite a difference. So, I need to add a pair of parentheses to make it equal to 479.

I need to think about where to place the parentheses to alter the order of operations to achieve the desired result.

Let me consider different possibilities:

1. Parenthesizing individual terms:

For example, (1 + 2) * 3 + 4 * 5 + 6 * 7 + 8 * 9

Calculate inside the parentheses first:

(1 + 2) = 3

Then, 3 * 3 = 9

Now, the equation becomes:

9 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 9 + 20 + 42 + 72 = 143

Still not 479.

2. Parenthesizing more terms:

What if I do (1 + 2 * 3) + 4 * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

1 + 2 * 3

Here, multiplication comes before addition, so 2 * 3 = 6, then 1 + 6 = 7

So, the equation becomes:

7 + 4 * 5 + 6 * 7 + 8 * 9

Again, following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 7 + 20 + 42 + 72 = 141

Same as before.

3. Let’s try (1 + 2) * (3 + 4) * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

(1 + 2) = 3

(3 + 4) = 7

So, 3 * 7 * 5 = 105

Now, the equation becomes:

105 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 105 + 42 + 72 = 219

Still not 479.

4. Maybe trying to group more terms:

(1 + 2 * 3 + 4) * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

1 + 2 * 3 + 4

Multiplication first: 2 * 3 = 6

Then, 1 + 6 + 4 = 11

So, 11 * 5 = 55

Now, the equation becomes:

55 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 55 + 42 + 72 = 169

Getting closer, but still not 479.

5. Let’s try a different approach. Maybe grouping the additions together:

1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)

First, calculate inside the parentheses:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 6 + 20 + 42 + 72 = 140

Then, 1 + 140 = 141

Same as before.

6. Perhaps grouping two multiplications together:

1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

(3 + 4) = 7

So, 2 * 7 * 5 = 70

Now, the equation becomes:

1 + 70 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 1 + 70 + 42 + 72 = 185

Still not 479.

7. Maybe grouping more multiplications:

1 + 2 * 3 + 4 * (5 + 6) * 7 + 8 * 9

First, calculate inside the parentheses:

(5 + 6) = 11

So, 4 * 11 * 7 = 308

Now, the equation becomes:

1 + 2 * 3 + 308 + 8 * 9

Following order of operations:

2 * 3 = 6

8 * 9 = 72

So, 1 + 6 + 308 + 72 = 387

Closer, but still not 479.

8. Let’s try grouping the last terms:

1 + 2 * 3 + 4 * 5 + 6 * (7 + 8) * 9

First, calculate inside the parentheses:

(7 + 8) = 15

So, 6 * 15 * 9 = 810

Now, the equation becomes:

1 + 2 * 3 + 4 * 5 + 810

Following order of operations:

2 * 3 = 6

4 * 5 = 20

So, 1 + 6 + 20 + 810 = 837

Now that’s way over 479.

Maybe I need to group terms in a way that subtraction is implied, but since there are no subtractions in the original equation, perhaps I need to consider that.

Wait, maybe the problem allows for adding parentheses to change the operations indirectly.

Alternatively, perhaps the parentheses need to group additions in a specific way.

Let me try grouping the first few terms:

(1 + 2 * 3 + 4 * 5) + 6 * 7 + 8 * 9

First, calculate inside the parentheses:

2 * 3 = 6

4 * 5 = 20

So, 1 + 6 + 20 = 27

Now, the equation becomes:

27 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 27 + 42 + 72 = 141

Same as before.

9. Maybe grouping further:

(1 + 2 * 3 + 4 * 5 + 6 * 7) + 8 * 9

First, calculate inside the parentheses:

2 * 3 = 6

4 * 5 = 20

6 * 7 = 42

So, 1 + 6 + 20 + 42 = 69

Now, the equation becomes:

69 + 8 * 9

Following order of operations:

8 * 9 = 72

So, 69 + 72 = 141

Still not 479.

10. Let’s try grouping differently:

1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)

First, calculate inside the parentheses:

3 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations inside:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 3 + 20 + 42 + 72 = 137

Now, 2 * 137 = 274

Then, the equation becomes:

1 + 274 = 275

Still not 479.

11. Maybe trying to group the additions and multiplications in a different way:

1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)

Wait, I did this earlier and got 141.

Hmm.

Let me think differently. Maybe I need to group terms to create a larger multiplication.

For example:

1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)

First, calculate inside the parentheses:

3 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 3 + 20 + 42 + 72 = 137

Now, 2 * 137 = 274

Then, 1 + 274 = 275

Still not 479.

12. Let’s try another approach:

(1 + 2) * (3 + 4) * (5 + 6) * (7 + 8) * 9

Wait, that’s adding multiple sets of parentheses, which is not allowed since I can only add one pair.

Back to the original problem: only one pair of parentheses can be added.

Maybe I need to group a subtraction or division, but those operations aren’t present in the original equation.

Wait, perhaps I need to consider that adding parentheses can create a negative term, but again, there are no subtractions in the original equation.

Alternatively, maybe the parentheses are meant to group additions in a specific way to create a larger number.

Let me try grouping more terms together:

1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)

First, inside the parentheses:

3 + 4 * 5 + 6 * 7 + 8 * 9

Following order of operations:

4 * 5 = 20

6 * 7 = 42

8 * 9 = 72

So, 3 + 20 + 42 + 72 = 137

Now, 2 * 137 = 274

Then, 1 + 274 = 275

Same as before.

13. Maybe trying to group the last few terms:

1 + 2 * 3 + 4 * 5 + 6 * (7 + 8 * 9)

First, inside the parentheses:

7 + 8 * 9

Following order of operations:

8 * 9 = 72

Then, 7 + 72 = 79

Now, 6 * 79 = 474

So, the equation becomes:

1 + 2 * 3 + 4 * 5 + 474

Following order of operations:

2 * 3 = 6

4 * 5 = 20

So, 1 + 6 + 20 + 474 = 501

Close to 479, but still not there.

14. Maybe adjusting the grouping slightly:

1 + 2 * 3 + 4 * (5 + 6 * 7 + 8 * 9)

First, inside the parentheses:

5 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 5 + 42 + 72 = 119

Now, 4 * 119 = 476

Then, the equation becomes:

1 + 2 * 3 + 476

Following order of operations:

2 * 3 = 6

So, 1 + 6 + 476 = 483

Still not 479.

15. Let’s try grouping the first part differently:

(1 + 2 * 3 + 4) * (5 + 6 * 7 + 8 * 9)

First, calculate inside the first set of parentheses:

1 + 2 * 3 + 4

Following order of operations:

2 * 3 = 6

Then, 1 + 6 + 4 = 11

Now, the second set of parentheses:

5 + 6 * 7 + 8 * 9

Following order of operations:

6 * 7 = 42

8 * 9 = 72

So, 5 + 42 + 72 = 119

Now, 11 * 119 = 1309

Way over 479.

16. Maybe trying to group only two terms:

1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9

Wait, I did this earlier and got 185.

Hmm.

Let me think differently. Maybe I need to group terms to create a division or subtraction, but since those operations aren’t present, perhaps I need to consider that the parentheses can be used to create a fraction or some other operation, but that might be reaching too far.

Alternatively, maybe the problem allows for adding a pair of parentheses around the entire equation, but that doesn’t make sense.

Wait, perhaps the parentheses are meant to group additions in a specific way to create a larger number.

Let me try grouping more terms together for multiplication.

For example:

1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9

First, inside the parentheses:

3 + 4 * 5 + 6

Following order of operations:

4 * 5 = 20

So, 3 + 20 + 6 = 29

Now, 2 * 29 * 7 = 406

Then, the equation becomes:

1 + 406 + 8 * 9

Following order of operations:

8 * 9 = 72

So, 1 + 406 + 72 = 479

Wait a minute, that’s exactly 479!

Let me double-check:

First, inside the parentheses: 3 + 4 * 5 + 6

4 * 5 = 20

Then, 3 + 20 + 6 = 29

Then, 2 * 29 * 7 = 2 * 203 = 406

Then, 1 + 406 = 407

Plus 8 * 9 = 72

So, 407 + 72 = 479

Perfect! So, by adding parentheses around (3 + 4 * 5 + 6), and multiplying that sum by 2 and 7, we get the desired result.

So, the corrected equation is:

1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9 = 479

Final Answer

$$ \boxed{1 + 2 \times (3 + 4 \times 5 + 6) \times 7 + 8 \times 9 = 479} $$

So I need to find the least odd prime factor of $ 2019^8 + 1 $. Hmm, that seems pretty big, but I think I can break it down step by step.

First, I know that finding prime factors of large exponents can be tricky, but there might be some properties or theorems that can help simplify the process. Since the expression is $ 2019^8 + 1 $, maybe I can look for patterns or ways to factor this expression.

I recall that expressions of the form $ a^n + b^n $ can sometimes be factored, especially when $ n $ is a power of 2. In this case, $ n = 8 $, which is $ 2^3 $, so that might be useful.

Let me see if I can factor $ 2019^8 + 1 $. I know that $ a^4 + 1 $ can be factored as $ (a^4 + 1) = (a^2 + \sqrt{2}a + 1)(a^2 - \sqrt{2}a + 1) $, but that involves irrational numbers, which isn’t helpful for factoring into integers. Maybe there’s another way.

Wait, perhaps I can write $ 2019^8 + 1 $ as $ (2019^4)^2 + 1^2 $, which resembles the sum of squares. I know that over the integers, sums of squares don’t factor nicely, but maybe there’s a different approach.

Another thought: maybe I can consider smaller exponents first and see if there’s a pattern or a recurring factor that appears.

Let’s try computing $ 2019^2 + 1 $, $ 2019^4 + 1 $, and so on, and see if I can find any common factors or patterns.

First, calculate $ 2019^2 + 1 $:

$ 2019^2 = 2019 \times 2019 $. I don’t need the exact value right now; maybe I can just keep it as $ 2019^2 $ for now.

So, $ 2019^2 + 1 = 2019^2 + 1 $. Doesn’t seem immediately helpful.

Next, $ 2019^4 + 1 $. Again, it’s similar to the previous expression, and I don’t see an obvious factorization.

Maybe I should try to see if there are any small prime numbers that divide $ 2019^8 + 1 $. Since I’m looking for the least odd prime factor, I can start testing small primes like 3, 5, 7, 11, and so on, to see if any of them divide $ 2019^8 + 1 $.

Let’s start with 3. To check if 3 divides $ 2019^8 + 1 $, I can compute $ 2019^8 + 1 $ modulo 3.

First, find $ 2019 $ modulo 3. Since $ 2019 = 3 \times 673 $, $ 2019 \equiv 0 \pmod{3} $. Therefore, $ 2019^8 \equiv 0^8 \equiv 0 \pmod{3} $, and $ 2019^8 + 1 \equiv 0 + 1 \equiv 1 \pmod{3} $. So, 3 does not divide $ 2019^8 + 1 $.

Next, check 5. Compute $ 2019 $ modulo 5. $ 2019 \div 5 = 403 $ with a remainder of 4, so $ 2019 \equiv 4 \pmod{5} $. Then, $ 2019^8 \equiv 4^8 \pmod{5} $. Since $ 4 \equiv -1 \pmod{5} $, $ 4^8 \equiv (-1)^8 \equiv 1 \pmod{5} $. Therefore, $ 2019^8 + 1 \equiv 1 + 1 \equiv 2 \pmod{5} $, so 5 does not divide $ 2019^8 + 1 $.

Now, check 7. Find $ 2019 $ modulo 7. $ 2019 \div 7 = 288 $ with a remainder of 3, so $ 2019 \equiv 3 \pmod{7} $. Then, $ 2019^8 \equiv 3^8 \pmod{7} $. Let’s compute $ 3^8 $ modulo 7.

First, find the pattern of powers of 3 modulo 7:

$ 3^1 \equiv 3 \pmod{7} $

$ 3^2 \equiv 9 \equiv 2 \pmod{7} $

$ 3^3 \equiv 6 \pmod{7} $

$ 3^4 \equiv 18 \equiv 4 \pmod{7} $

$ 3^5 \equiv 12 \equiv 5 \pmod{7} $

$ 3^6 \equiv 15 \equiv 1 \pmod{7} $

Since $ 3^6 \equiv 1 \pmod{7} $, then $ 3^8 = 3^{6+2} = 3^6 \times 3^2 \equiv 1 \times 2 \equiv 2 \pmod{7} $. Therefore, $ 2019^8 + 1 \equiv 2 + 1 \equiv 3 \pmod{7} $, so 7 does not divide $ 2019^8 + 1 $.

Next, check 11. Compute $ 2019 $ modulo 11. $ 2019 \div 11 = 183 $ with a remainder of 6, so $ 2019 \equiv 6 \pmod{11} $. Then, $ 2019^8 \equiv 6^8 \pmod{11} $. Let’s find a pattern for powers of 6 modulo 11:

$ 6^1 \equiv 6 \pmod{11} $

$ 6^2 \equiv 36 \equiv 3 \pmod{11} $

$ 6^3 \equiv 18 \equiv 7 \pmod{11} $

$ 6^4 \equiv 42 \equiv 9 \pmod{11} $

$ 6^5 \equiv 54 \equiv 10 \pmod{11} $

$ 6^6 \equiv 60 \equiv 5 \pmod{11} $

$ 6^7 \equiv 30 \equiv 8 \pmod{11} $

$ 6^8 \equiv 48 \equiv 4 \pmod{11} $

Therefore, $ 2019^8 + 1 \equiv 4 + 1 \equiv 5 \pmod{11} $, so 11 does not divide $ 2019^8 + 1 $.

Continuing this process seems tedious. Maybe there’s a better way to approach this problem.

Let me think differently. Perhaps I can consider the expression $ 2019^8 + 1 $ and see if it can be expressed in a form that reveals its prime factors.

I recall that for expressions like $ a^4 + 4b^4 $, there’s a Sophie Germain identity: $ a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2) $. Maybe I can manipulate $ 2019^8 + 1 $ into a similar form.

Wait, $ 2019^8 + 1 = (2019^4)^2 + 1^2 $. This resembles the sum of squares, but over integers, sums of squares don’t factor nicely. Maybe I can consider complex numbers or some other algebraic identities, but that might not help in finding integer prime factors.

Alternatively, perhaps I can consider modulo properties to find possible prime factors. For example, if $ p $ is an odd prime that divides $ 2019^8 + 1 $, then $ 2019^8 \equiv -1 \pmod{p} $, which implies $ 2019^{16} \equiv 1 \pmod{p} $. This means that the order of 2019 modulo $ p $ divides 16, and since $ 2019^8 \equiv -1 \pmod{p} $, the order can’t divide 8. Therefore, the order must be 16.

By Fermat’s Little Theorem, the order of 2019 modulo $ p $ divides $ p - 1 $. Therefore, $ p - 1 $ must be divisible by 16. So, $ p \equiv 1 \pmod{16} $.

This narrows down the possible primes $ p $ that could divide $ 2019^8 + 1 $ to those primes congruent to 1 modulo 16.

Let me list some primes that are 1 modulo 16: 17, 97, 113, 193, 241, 257, etc.

Now, I can test these primes to see if any of them divide $ 2019^8 + 1 $.

Start with 17. Compute $ 2019 $ modulo 17.

First, find $ 2019 \div 17 $. 17 times 118 is 2006, so $ 2019 - 2006 = 13 $. Therefore, $ 2019 \equiv 13 \pmod{17} $.

Then, $ 2019^8 \equiv 13^8 \pmod{17} $.

Compute $ 13^2 \equiv 169 \equiv 16 \pmod{17} $

$ 13^4 = (13^2)^2 \equiv 16^2 \equiv 256 \equiv 1 \pmod{17} $

$ 13^8 = (13^4)^2 \equiv 1^2 \equiv 1 \pmod{17} $

Therefore, $ 2019^8 + 1 \equiv 1 + 1 \equiv 2 \pmod{17} $, so 17 does not divide $ 2019^8 + 1 $.

Next, check 97. Compute $ 2019 $ modulo 97.

First, find how many times 97 fits into 2019. 97 times 20 is 1940, and 2019 - 1940 = 79. So, $ 2019 \equiv 79 \pmod{97} $.

Then, $ 2019^8 \equiv 79^8 \pmod{97} $.

This seems a bit cumbersome to compute directly. Maybe I can find a pattern or use Fermat’s Little Theorem, which states that $ a^{p-1} \equiv 1 \pmod{p} $ if $ p $ is prime and $ p $ does not divide $ a $.

Here, $ p = 97 $, so $ 79^{96} \equiv 1 \pmod{97} $. But I need $ 79^8 $, which is not directly helpful. Maybe I can compute $ 79^2 $, then $ 79^4 $, and so on.

First, compute $ 79^2 \pmod{97} $:

$ 79^2 = 6241 $. Now, divide 6241 by 97.

97 times 64 is 6208, so 6241 - 6208 = 33. Therefore, $ 79^2 \equiv 33 \pmod{97} $.

Next, $ 79^4 = (79^2)^2 \equiv 33^2 \pmod{97} $.

$ 33^2 = 1089 $. 97 times 11 is 1067, so 1089 - 1067 = 22. Therefore, $ 79^4 \equiv 22 \pmod{97} $.

Then, $ 79^8 = (79^4)^2 \equiv 22^2 \pmod{97} $.

$ 22^2 = 484 $. 97 times 4 is 388, so 484 - 388 = 96. Therefore, $ 79^8 \equiv 96 \pmod{97} $.

Thus, $ 2019^8 + 1 \equiv 96 + 1 \equiv 97 \equiv 0 \pmod{97} $.

So, 97 divides $ 2019^8 + 1 $. Since 97 is the smallest prime congruent to 1 modulo 16 that we’ve checked and it divides $ 2019^8 + 1 $, it must be the least odd prime factor of $ 2019^8 + 1 $.

Therefore, the least odd prime factor is 97.

Final Answer

$$ \boxed{97} $$