What is the product of the largest and the smallest possible values of m for which a number 5m 83m 4m1 is divisible by 9?a)16b)40c)80d)10Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Analysis:

To find the product of the largest and smallest possible values of m for which the number 5m 83m 4m1 is divisible by 9, we need to understand the divisibility rule for 9. A number is divisible by 9 if the sum of its digits is divisible by 9.

Solution:

1. The number given is 5m 83m 4m1.
2. Sum of the digits in the number = 5 + m + 8 + 3 + m + 4 + m + 1 = 21 + 3m.
3. For the number to be divisible by 9, the sum of the digits must also be divisible by 9.
4. Therefore, 21 + 3m must be divisible by 9.
5. The smallest value of m that satisfies this condition is m = 3 (21 + 3*3 = 30, which is divisible by 9).
6. The largest value of m that satisfies this condition is m = 6 (21 + 3*6 = 39, which is divisible by 9).
7. The product of the largest and smallest possible values of m = 3 * 6 = 18.

Therefore, the product of the largest and smallest possible values of m for which the number 5m 83m 4m1 is divisible by 9 is 18, which corresponds to option A.

Answer

Concept:

Divisibility rule of 9: A number is divisible by 9, when the sum of the digits of the number is divisible by 9.

Calculation:

5 + m + 8 + 3 + m + 4 + m + 1

for minimum value

⇒ 21 + 3m

⇒ Minimum value of m = 2 (as 27 is the multiple of 9)

For maximum value

⇒ 21 + 3m (as 45 is the multiple of 9)

⇒ Maximum value of m = 8

∴ The product of largest and smallest value

⇒ 8 × 2 = 16

Key Points

Here m is a digit (m can be 0, 1, 2, 3,.....,8) not other than this.

We can't take m = 20, m = 30 etc.

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