How many 5 digit numbers can be formed using the digits 1, 2, 3, 4, 5

Question

How many 5 digit numbers can be formed using the digits 1, 2, 3, 4, 5 and 6 (without repetition) that are divisible by 8?

(A) 56
(B) 64
(C) 72
(D) 84
(E) 96

GMATWhizTeam · Answer

Solution   :

We are given,
 • The digits 1, 2, 3, 4, 5 and 6 
We need to form 5 digit numbers that are divisible by 8 using the given 6 digits ( without repetition ).

We know for a number to be divisible by 8,  the last 3 digits of the number must be divisible by 8 . 
All the possible cases when the last 3 digits are divisible by 8 are :
1. 136
2. 152
3. 216
4. 256
5. 264
6. 312
7. 352
8. 416
9. 432
10. 456
11. 512
12. 536
13. 624
14. 632

Suppose if we consider Case 1 i.e., the number having the last 3 digits as 136, then we have :
    _ _  1   3   6     
 • The 1st place can be filled with the remaining 3 digits (2, 4, 5) in  3 ways 
• The 2nd place can be filled in  2 ways  (after filling 1st place with any one of the digits 2, 4 or 5, we will be left with 2 digits)
• So the total number of ways =  3 * 2 = 6 ways

However, this is just for one particular case.
For  14 such cases , the total number of ways =  6 * 14  =  84 ways

Hence, the correct answer is   Option D  .

pudu · Answer

It is taking almost 5 minutes. Is there any way to solve it quickly?

kingbucky · Answer

To find all the 3 digit multiples of 8 with those numbers, list out all possible unit digits (must be divisible by 2):
2
4
6

Now, let's find the possible last 2 digts (must be divisible by 4):
12
16
24
32
36
52
56
64

Now, let's find the possible 3 digit numbers divisible by 8:
Note, these numbers should have last two digits divisible by 4

Now, lets append 1 to 
12
16
24
32
36
52
56
64
and start checking

Is 112 divisible by 8 ?
let divible by 4 first, we have 100/4 +12/4 = 25+3 (odd +odd) = even ; ie, has another 2 as factor, hence divisible by 8
if 1 is appended before, we must have 25(odd)+ (last 2 digits /4) odd = even if we want it to be divisible by 8.
List out numbers of the form odd *4 from our list,

similarly for digit 2 appended before, we will have 50(even)+even =even;
List out numbers of the form even*4 from our list,

By following this, you can quickly get all possible multiples of 8 using the digits 1 to 6.

You can then choose 2 numbers from 3 and arrange them to get the final answer.

How many 5 digit numbers can be formed using the digits 1, 2, 3, 4, 5

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GMAT Problem Solving (PS) Questions

How many 5 digit numbers can be formed using the digits 1, 2, 3, 4, 5

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