If I multiply all whole numbers from 1 through 10 what is the largest power of 4 that is a factor of the product?

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08-02-2017
Mathematics

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If I multiply all whole numbers from 1 through 10 what is the largest power of 4 that is a factor of the product?

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taskmasters taskmasters · Answer 1 · 2017-02-10T13:35:20+01:00

If we multiply 1 through 10, the mathematical representation of the product would be,
P = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10
Getting the factors of each number,
P = 1 x 2 x 3 x (2 x 2) x 5 x (3 x 2) x 7 x (2 x 2 x 2) x 9 x (5 x 2)
Then, we take out all the 2's.
2 x (2 x 2) x (3 x 2) x (2 x 2 x 2) x (5 x 2)
Combining all the 2's.
2^8 x 3 x 5
2^8 can also be written as (2 x 2)^4. Thus, the answer is 4.

MrRoyal MrRoyal · Answer 2 · 2021-10-22T12:33:58+02:00

Powers and factors are both used to group numbers and expressions.

The largest power of 4 that is a factor of the product is 12

The product is represented as:

[tex]\mathbf{Product = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10}[/tex]

Expand

[tex]\mathbf{Product = 1 \times 2 \times 3 \times 2 \times 2 \times 5 \times 2 \times 3 \times 7 \times 2 \times 2 \times 2 \times 3 \times 3 \times 2 \times 5}[/tex]

Rewrite as:

[tex]\mathbf{Product = 1 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 3 \times 7 \times 3 \times 3 \times 5}[/tex]

Rewrite as a product of 4

[tex]\mathbf{Product = 1 \times 4 \times 4 \times 4 \times 4 \times 3 \times 5 \times 3 \times 7 \times 3 \times 3 \times 5}[/tex]

Express as powers of 4

[tex]\mathbf{Product = 1 \times 4^4 \times 3 \times 5 \times 3 \times 7 \times 3 \times 3 \times 5}[/tex]

Regroup as:

[tex]\mathbf{Product = 1 \times 4^4 \times 3 \times 3 \times 3 \times 3 \times 7 \times 5 \times 5}[/tex]

Express as a power of 4

[tex]\mathbf{Product = 1 \times 4^4 \times 3^4 \times 7 \times 5 \times 5}[/tex]

Apply law of indices

[tex]\mathbf{Product = 1 \times (4 \times 3)^4 \times 7 \times 5 \times 5}[/tex]

[tex]\mathbf{Product = 1 \times (12)^4 \times 7 \times 5 \times 5}[/tex]

The above means that, the largest power of 4 that is a factor of the product is 12

Read more about factors and powers at:

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