abc is a three digit number where a, b, c are their digits. How many are there such that a×b×c =12
Answers
Answer:
15 three digit numbers
Step-by-step explanation:
Let me write the factors of 12.
1, 2, 3, 4, 6, 12.
We can’t take 12 as either a, b or c because 12 is a two digit number and digits are only of one digit numbers.
‘The following is the no. of numbers without digit repetition.’
Let one digit be 1. So the product of other digits will be 12 / 1 = 12.
We can either take the two digits as 2, 6 or as 3, 4.
No. of three digit numbers formed by 1, 2, 6 is 3! = 6.
No. of three digit numbers formed by 1, 3, 4 is 3! = 6.
When one digit is let as 2, then the product of the others will be 12 / 2 = 6.
We can take the others as 1, 6. But it’s calculated. When it’s taken as 2, 3, digit repetition occurs. So there’s nothing to calculate. So is 3, 4 and 6.
Hence there are a total of 6 + 6 = 12 numbers can be formed (without digit repetition).
The following is that with digit repetition.
12 numbers were found with one digit as 1.
When one digit is let as 2, the others will be 2 and 3. (Nos. of digits 2, 1, 6 are found.)
No. of three digit numbers formed by 2, 2, 3 is 3! / 2! = 3.
When 3 is let as one digit, the product of the others is 12 / 3 = 4.
3 with others as 1 and 4 is calculated earlier. So is 3, 2, 2.
One digit with either 4 or 6 is also found. ‘
So there are a total of 12 + 3 = 15 three digit numbers
There are 15 numbers that can be formed using the given conditions.
Given:
abc is a 3-digit number
a × b × c = 12
To Find:
how many values of abc exist
Solution:
the prime factorization of 12 is given as-
12 = 3 × 2 × 2 × 1
Now, the factor of 12 are- 1, 2, 3, 4, 6, 12
since a, b, and c are digits of a number, they cannot take the value 12.
Thus, a, b, and c can take values 1, 2, 3, 4, and 6
when a = 1, abc can be 126, 134, 143, 162
when a = 2, abc can be 216, 223, 232, 261
when a = 3, abc can be 314, 322, 341
when a = 4, abc can be 413, 431
when a = 6, abc can be 612, 621
Adding all the numbers we get a total of 15 numbers
Thus, there are 15 numbers that can be formed using the given conditions.
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