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Math Expert V
Joined: 02 Sep 2009
Posts: 59144
The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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1
8 00:00

Difficulty:   45% (medium)

Question Stats: 66% (01:31) correct 34% (01:31) wrong based on 376 sessions

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Tough and Tricky questions: Number Properties.

The "length" of a positive integer greater than 1 is the number of prime numbers, including repeats, that are factors of that integer. For instance, the length of 20 is 3, because $$2 \times 2 \times 5 = 20$$. What is the length of 5,950?

A. 1
B. 2
C. 3
D. 4
E. 5

Kudos for a correct solution.

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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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3
2
5950=2*5*595
=2*5*5*119
=2*5*5*17*7

so , length is 5

##### General Discussion
Manager  Joined: 10 Sep 2014
Posts: 96
Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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2
5950
=5x1190
=5x5x238
=5x5x2x118
=5x5x2x2x59

5 prime factors including repeats

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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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4

$$5950 = 2*5*595$$

$$= 2 * 5 * 5 * 119$$

$$= 2 * 5 * 5 * 7 * 17$$
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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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DangerPenguin wrote:
5950
=5x1190
=5x5x238
=5x5x2x118
=5x5x2x2x59

5 prime factors including repeats

Seems to be typo; though answer is correct _________________
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Posts: 59144
Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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1
Official Solution:

The "length" of a positive integer greater than 1 is the number of prime numbers, including repeats, that are factors of that integer. For instance, the length of 20 is 3, because $$2 \times 2 \times 5 = 20$$. What is the length of 5,950?

A. 1
B. 2
C. 3
D. 4
E. 5

In this problem, we should notice that the "length" of an integer is defined in some way involving prime factors. Therefore, even if we are not 100% sure what "length" means, we should factor 5,950 down to its prime factors and examine what we get.

First, it's easy to take out a factor of 10: $$5950 = 10 \times 595$$.

10 is very simple to factor: $$10 = 2 \times 5$$.

Since 595 ends in 5, we know that 595 is divisible by 5. Actually performing this division, we wind up with $$595 = 5 \times 119$$.

So $$5950 = 2 \times 5 \times 5 \times 119$$.

Now, 119 "looks" prime, but you must test it with primes up to the square root of 119, which is approximately 11. (If 119 is not prime, then at least one of its prime factors must be smaller than the square root of 119.)

It turns out that 119 is divisible by 7. $$119 = 7 \times 17$$.

Thus, we now have the full prime factorization of 5,950: $$5950 = 2 \times 5 \times 5 \times 7 \times 17$$.

Finally, we return to the definition of "length." We are given the example that the length of 20 is 3, since $$2 \times 2 \times 5 = 20$$. So we can see that "length" is just the number of prime numbers in the prime factorization, counting repeats (such as 2 in the example of 20).

Thus, the length of 5,950 is 5.

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Math Expert V
Joined: 02 Sep 2009
Posts: 59144
Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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Bunuel wrote:
Official Solution:

The "length" of a positive integer greater than 1 is the number of prime numbers, including repeats, that are factors of that integer. For instance, the length of 20 is 3, because $$2 \times 2 \times 5 = 20$$. What is the length of 5,950?

A. 1
B. 2
C. 3
D. 4
E. 5

In this problem, we should notice that the "length" of an integer is defined in some way involving prime factors. Therefore, even if we are not 100% sure what "length" means, we should factor 5,950 down to its prime factors and examine what we get.

First, it's easy to take out a factor of 10: $$5950 = 10 \times 595$$.

10 is very simple to factor: $$10 = 2 \times 5$$.

Since 595 ends in 5, we know that 595 is divisible by 5. Actually performing this division, we wind up with $$595 = 5 \times 119$$.

So $$5950 = 2 \times 5 \times 5 \times 119$$.

Now, 119 "looks" prime, but you must test it with primes up to the square root of 119, which is approximately 11. (If 119 is not prime, then at least one of its prime factors must be smaller than the square root of 119.)

It turns out that 119 is divisible by 7. $$119 = 7 \times 17$$.

Thus, we now have the full prime factorization of 5,950: $$5950 = 2 \times 5 \times 5 \times 7 \times 17$$.

Finally, we return to the definition of "length." We are given the example that the length of 20 is 3, since $$2 \times 2 \times 5 = 20$$. So we can see that "length" is just the number of prime numbers in the prime factorization, counting repeats (such as 2 in the example of 20).

Thus, the length of 5,950 is 5.

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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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5950=595*10= 2*5^2*7*17

5950 has four distinct prime factors BUT 5 is the length of 5950 because we have to consider repeats without writing the number in the exponential form.
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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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5950/2= 2975
2975/5=595
595/5=123
123/3=31

Therefore 2x5x5x3x31=5950

And the length is 5

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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Number Properties.

The "length" of a positive integer greater than 1 is the number of prime numbers, including repeats, that are factors of that integer. For instance, the length of 20 is 3, because $$2 \times 2 \times 5 = 20$$. What is the length of 5,950?

A. 1
B. 2
C. 3
D. 4
E. 5

Kudos for a correct solution.

5,950/50=119
(50)*(119)=(2*5*5)*(7*17)
5
E
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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Number Properties.

The "length" of a positive integer greater than 1 is the number of prime numbers, including repeats, that are factors of that integer. For instance, the length of 20 is 3, because $$2 \times 2 \times 5 = 20$$. What is the length of 5,950?

A. 1
B. 2
C. 3
D. 4
E. 5

Let’s break 5,950 into prime factors.

5,950 = 595 x 10 = 119 x 5 x 5 x 2 = 7 x 17 x 5 x 5 x 2

Thus, the length is 5.

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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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Bunuel wrote:

Tough and Tricky questions: Number Properties.

The "length" of a positive integer greater than 1 is the number of prime numbers, including repeats, that are factors of that integer. For instance, the length of 20 is 3, because $$2 \times 2 \times 5 = 20$$. What is the length of 5,950?

A. 1
B. 2
C. 3
D. 4
E. 5

Kudos for a correct solution.

5950 = 2*5*5*7*17

119 tricked me for a while.

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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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5950 = 5*5*2*7*17 all primes (Taking L.C.M )
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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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1
5950=2*5*5*7*17, the lenght is 5, E
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The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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[quote="Bunuel"]

Tough and Tricky questions: Number Properties.

The "length" of a positive integer greater than 1 is the number of prime numbers, including repeats, that are factors of that integer. For instance, the length of 20 is 3, because $$2 \times 2 \times 5 = 20$$. What is the length of 5,950?

A. 1
B. 2
C. 3
D. 4
E. 5

5950=2*5*5*7*17
so ans.E
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Re: The "length" of a positive integer greater than 1 is the number of pri  [#permalink]

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_________________ Re: The "length" of a positive integer greater than 1 is the number of pri   [#permalink] 18 Jul 2019, 22:35
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