Three distinct prime numbers, less than 10 are taken and all the numbers that can be formed by arranging all the digits are taken. Now, difference between the largest and the smallest number formed is equal to 495. It is also given that sum of the digits is more than 13. What is the product of the numbers?

a)
30

b)
70

c)
105

d)
315

Correct answer is option 'B'. Can you explain this answer?

Question

Three distinct prime numbers, less than 10 are taken and all the numbers that can be formed by arranging all the digits are taken. Now, difference between the largest and the smallest number formed is equal to 495. It is also given that sum of the digits is more than 13. What is the product of the numbers?

a)
30
b)
70
c)
105
d)
315

Correct answer is option 'B'. Can you explain this answer?

Accepted Answer

Prime numbers less than 10 = 2, 3, 5, 7 So, option (d) cannot be the answer.
Sum of digits is more than 13 - so, set of [2, 3, 5] is not possible.
So, option (a) cannot be the answer.
Now check for options (b) and (c) by taking the values.

Answer

Since the sum of digits greater than 13...
two possible sets of prine no less than 10
(3 5 7) or (2 5 7 ) apply 2nd cond giving difference of largest and snalles number (752- 257=495)
7* 5*2 = 70

Answer

When we take 3+5+7=15
which implies that it is greater than 13

so the product will be 3*5*7*=105
option C is the correct answer

Answer

Solution:

Given that three distinct prime numbers, less than 10 are taken and all the numbers that can be formed by arranging all the digits are taken.

We know that prime numbers less than 10 are 2, 3, 5, and 7.

So, out of these four numbers, we can select three prime numbers in 4C3 ways = 4 ways.

They are (2, 3, 5), (2, 3, 7), (2, 5, 7), and (3, 5, 7).

1. Calculation of the largest and smallest number formed:
Let us take the first set (2, 3, 5) and find out the numbers that can be formed by arranging these digits.

The possible numbers are 235, 253, 325, 352, 523, and 532.

The largest number formed is 532 and the smallest number formed is 235.

The difference between the largest and smallest numbers is 532 - 235 = 297.

Let us analyze the other three sets:

For the second set (2, 3, 7), the largest number formed is 732 and the smallest number formed is 237.

The difference between the largest and smallest numbers is 732 - 237 = 495.

For the third set (2, 5, 7), the largest number formed is 752 and the smallest number formed is 257.

The difference between the largest and smallest numbers is 752 - 257 = 495.

For the fourth set (3, 5, 7), the largest number formed is 753 and the smallest number formed is 357.

The difference between the largest and smallest numbers is 753 - 357 = 396.

Hence, the only set that satisfies the given condition is (2, 3, 7).

2. Calculation of the sum of digits:
The sum of digits in the numbers formed using the digits 2, 3, and 7 are:

2 + 3 + 7 = 12
2 + 7 + 3 = 12
3 + 2 + 7 = 12
3 + 7 + 2 = 12
7 + 2 + 3 = 12
7 + 3 + 2 = 12

The sum of the digits is always 12, which is less than 13.

So, the given condition is not satisfied.

Hence, there is no set that satisfies both the given conditions.

Therefore, the given problem has no solution.

The correct option is (D) None of these.

Note: The answer key given in the question is incorrect.

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