If the third digit of a three digit positive integer equals the

Question

If the third digit of a three digit positive integer equals the product of the first two digits, how many such three digit positive integers are there?

A. 23
B. 24
C. 28
D. 32
E. 36

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Sudhir25 · Answer

The three digit no. such as 632,111,221,422 where the product of first two digits is equal to third digit.
We have to calculate such numbers.
We know that third digit of a three digit no. can be either 1,2,3,4,5,6,7,8,or 9.
So, our product must be equal to these nine digits.
For example, 111,221,212,422,414,441....
This is possible in 23 ways. Hence, A is our answer.

Dhimanghosh1234 · Answer

Alternate method: The brutal way:

1*1=1
1*2=2
1*3=3
.
.
1*9=9
Total  9  ways with 1 at 100th place.

2*1=2
2*2=4
2*3=6
2*4=8
Total  4  ways with 2 at 100th place

3*1=3
3*2=6
3*3=9
Total  3  ways
4*1=4
4*2=8
total  2  ways
For 5,6,7,8 and 9 there are 5*1= 5  ways

Total ways=9+4+3+2+5= 23  ways

IMO option A

unraveled · Answer

If the third digit of a three digit positive integer equals the product of the first two digits, how many such three digit positive integers are there?

A. 23
B. 24
C. 28
D. 32
E. 36

ABC is the integer where
C = A * B

Since C is a digit max value that it can take is 9 and least is 0. Hence, possibilities are as follows
9 = 1*9 = 9*1 = 3*3  (3 ways)   
8 = 1*8 = 8*1 = 2*4 = 4*2  (4 ways)   
7 = 1*7 = 7*1  (2 ways)   
6 = 1*6 = 6*1 = 2*3 = 3*2  (4 ways)   
5 = 1*5 = 5*1  (2 ways)   
4 = 1*4 = 4*1 = 2*2  (3 ways)   
3 = 1*3 = 3*1  (2 ways)   
2 = 1*2 = 2*1  (2 ways)   
1 = 1*1  (1 way)   
0 = 1*0 = 2*0 = 3*0 = .... = 9*0  (9 ways)

Here the factors are the digits that are used at the three positions.

NOTE: Better way is to calculate numbers in 100 series then 200 series and so on.. . There is a pattern.

Total ways = 3*2 + 4*2 + 2*4 + 1 + 9 = 32

Answer D.

harshtandel838 · Answer

The answer should be  (D) 32

We can solve it by considering the first digit and varying the second digit and finally checking whether the third digit can be considered a product of first two digits.

For n, n *   =   
For 1, 1 *   =  , Total range   =10
For 2, 2 *   =  , Total range = 5
For 3, 3 *   =  , Total range = 4
For 4, 4 *   =  , Total range = 3
For 5, 5 *   =  , Total range = 2
For 6, 6 *   =  , Total range = 2
For 7, 7 *   =  , Total range = 2
For 8, 8 *   =  , Total range = 2
For 9, 9 *   =  , Total range = 2

Sum of all total range = 32

If the third digit of a three digit positive integer equals the

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If the third digit of a three digit positive integer equals the

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