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# S99-17

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Math Expert
Joined: 02 Sep 2009
Posts: 65335

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16 Sep 2014, 00:53
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Difficulty:

65% (hard)

Question Stats:

56% (01:54) correct 44% (02:15) wrong based on 77 sessions

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The product of the digits of the four-digit number $$h$$ is 36. No two digits of $$h$$ are identical. How many different numbers are possible values of $$h$$?

A. 6
B. 12
C. 24
D. 36
E. 48

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Math Expert
Joined: 02 Sep 2009
Posts: 65335

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16 Sep 2014, 00:53
Official Solution:

The product of the digits of the four-digit number $$h$$ is 36. No two digits of $$h$$ are identical. How many different numbers are possible values of $$h$$?

A. 6
B. 12
C. 24
D. 36
E. 48

We start with the prime factorization of $$h$$, which is $$2^23^2$$. In other words, $$h$$ is the product of two 2's and two 3's. All the factors of 36, except for 1, can be constructed as products of some or all of these 2's and 3's: 2, 3, 4, 6, 9, 12, 18, and 36.

At this stage, we can see that the possible values of any digit of h only include 1, 2, 3, 4, 6, and 9. Any other digit would contain the wrong primes (5, 7) or too many 2's (8), or it would turn the product to zero (0).

We now test subgroups of four digits, imposing the conditions that the product of all four digits is 36 and that no two digits are the same. Start with one digit at a time; do not worry about the order of the digits yet.

9: If one of the digits is 9, then the product of the other three digits must be 4. The only possible sets of positive digits that have 3 members and multiply together to 4 are {2, 2, 1} and {4, 1, 1}, but these sets fail the condition that no two digits can be identical. Thus, 9 cannot be one of the digits of $$h$$.

6: If one of the digits is 6, then the product of the other three digits must be 6. The only possible set of positive digits that have 3 non-identical members and multiply together to 6 is {3, 2, 1}, again ignoring order. Thus, one possible set of the digits of $$h$$ is {6, 3, 2, 1}.

4: If one of the digits is 4, then the product of the other three digits must be 9. The only possible sets of positive digits that have 3 members and multiply together to 9 are {3, 3, 1} and {9, 1, 1}, but these sets fail the condition that no two digits can be identical. Thus, 4 cannot be one of the digits of $$h$$.

If we examine the remaining digits 3, 2, and 1, we see that they can only be part of the unordered set {6, 3, 2, 1}, if we are to satisfy all the given conditions.

Thus, the possible values of $$h$$ result from the rearrangement of these four digits. Since all the digits are distinct, the number of different rearrangements is simply $$4!$$, or $$(4)(3)(2)(1)$$, which equals 24.

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Senior Manager
Status: Math is psycho-logical
Joined: 07 Apr 2014
Posts: 393
Location: Netherlands
GMAT Date: 02-11-2015
WE: Psychology and Counseling (Other)

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27 Jan 2015, 04:18
Hi,

What I don't understand here is in this sentence "At this stage, we can see that the possible values of any digit of h only include 1, 2, 3, 4, 6, and 9. Any other digit would contain the wrong primes (5, 7) or too many 2's (8), or it would turn the product to zero (0)", why do we exclude 12 and 18? They have either 3,2,2 or 2,3,3 as their prime factors. Is it because they are already too large to be multiplied by any of the remaining factors and still yield 36?
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Joined: 02 May 2013
Posts: 7

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19 Oct 2015, 21:11
pacifist85 wrote:
Hi,

What I don't understand here is in this sentence "At this stage, we can see that the possible values of any digit of h only include 1, 2, 3, 4, 6, and 9. Any other digit would contain the wrong primes (5, 7) or too many 2's (8), or it would turn the product to zero (0)", why do we exclude 12 and 18? They have either 3,2,2 or 2,3,3 as their prime factors. Is it because they are already too large to be multiplied by any of the remaining factors and still yield 36?

Hi,

Bcoz, as you told 12 and 18 has prime factors 3,2,2 and 2,3,3, as per the question condition the product of distinct numbers to make 36 should not have identical numbers.
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"When the going gets tough, the tough get going"
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Joined: 12 Mar 2019
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25 Jul 2019, 04:43
why can't be just see the possible solution to make 36 with different no. I came with 2 such 4 digit only, 1236 and 0149
1236 will give us 4! to arrange these no. so 24 possible numbers :
0149 ; will lead to zero is multiplied:
So only solution is 24
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Joined: 28 May 2020
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29 May 2020, 12:12
Forget everything else written, the question just boils down to how many 4 digit numbers can you make without repeating 4 numbers ( and none can be 0)
Re: S99-17   [#permalink] 29 May 2020, 12:12

# S99-17

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