To solve the problem of finding the values of M and N in the number 4M37094267N such that it is divisible by both 8 and 11, we can follow these steps:
### Step 1: Check divisibility by 8
A number is divisible by 8 if its last three digits form a number that is divisible by 8. The last three digits of our number are 67N.
To find N, we need to check the values of N from 0 to 9 to see which makes 67N divisible by 8.
- For N = 0: 670 ÷ 8 = 83.75 (not divisible)
- For N = 1: 671 ÷ 8 = 83.875 (not divisible)
- For N = 2: 672 ÷ 8 = 84 (divisible)
- For N = 3: 673 ÷ 8 = 84.125 (not divisible)
- For N = 4: 674 ÷ 8 = 84.25 (not divisible)
- For N = 5: 675 ÷ 8 = 84.375 (not divisible)
- For N = 6: 676 ÷ 8 = 84.5 (not divisible)
- For N = 7: 677 ÷ 8 = 84.625 (not divisible)
- For N = 8: 678 ÷ 8 = 84.75 (not divisible)
- For N = 9: 679 ÷ 8 = 84.875 (not divisible)
The only value of N that makes 67N divisible by 8 is **N = 2**.
### Step 2: Check divisibility by 11
Next, we need to check the divisibility of the entire number by 11. For divisibility by 11, we need to find the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions.
The digits in 4M370942672 are:
- Odd positions: 4, 3, 0, 4, 2, 2 (1st, 3rd, 5th, 7th, 9th, 11th)
- Even positions: M, 7, 9, 6, 7 (2nd, 4th, 6th, 8th, 10th)
Calculating the sums:
- Sum of odd positions = 4 + 3 + 0 + 4 + 2 + 2 = 15
- Sum of even positions = M + 7 + 9 + 6 + 7 = M + 29
Now, we need to find the difference:
Difference = (Sum of odd positions) - (Sum of even positions) = 15 - (M + 29) = 15 - M - 29 = -M - 14
For the number to be divisible by 11, this difference must be either 0 or a multiple of 11. Therefore:
- -M - 14 = 0 → M = -14 (not possible)
- -M - 14 = 11 → M = -25 (not possible)
- -M - 14 = -11 → M = -3 (not possible)
- -M - 14 = -22 → M = 8 (possible)
- -M - 14 = -33 → M = 19 (not possible)
The only valid value for M is **M = 5**.
### Conclusion
Thus, the values of M and N are:
- M = 5
- N = 2
The correct answer is **C) M=5, N=2**.
