The five-digit number 45yz0 is divisible by 40. What is the maximum possible value of (y + z)?

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Accepted Answer

Step 1: The number is 45yz0. Since it ends with 0, it is divisible by 10.

Step 2: For divisibility by 40, the number must be divisible by both 8 and 5 (since 40 = 8 × 5).

Step 3: Divisibility by 5 is already satisfied because the last digit is 0.

Step 4: Check divisibility by 8. A number is divisible by 8 if its last three digits form a number divisible by 8.

Step 5: The last three digits are y, z, and 0, which form the number 100y + 10z + 0 = 100y + 10z.

Step 6: Since 100y + 10z = 10(10y + z), for divisibility by 8, 10(10y + z) must be divisible by 8.

Step 7: 10 mod 8 = 2, so the condition reduces to 2 × (10y + z) ≡ 0 (mod 8), or (10y + z) ≡ 0 (mod 4) because 2 × (10y + z) divisible by 8 means (10y + z) divisible by 4.

Step 8: So, (10y + z) must be divisible by 4.

Step 9: y and z are digits (0 to 9). We want to maximize y + z under the constraint that (10y + z) is divisible by 4.

Step 10: Try maximum y = 9, then find z such that (90 + z) divisible by 4.

90 mod 4 = 2 (since 4 × 22 = 88, remainder 2).

So, (90 + z) mod 4 = (2 + z) mod 4 = 0.

Therefore, z mod 4 = 2.

Possible z values with z mod 4 = 2 are 2, 6.

Check sums:
- y=9, z=6 → sum = 15
- y=9, z=2 → sum = 11

Step 11: Try y=8:
80 mod 4 = 0
So, (80 + z) mod 4 = z mod 4 = 0
Possible z values: 0, 4, 8
Sums:
- 8 + 8 = 16
- 8 + 4 = 12
- 8 + 0 = 8

Step 12: Try y=7:
70 mod 4 = 2
So, z mod 4 = 2 → z = 2 or 6
Sums:
- 7 + 6 = 13
- 7 + 2 = 9

Step 13: Try y=6:
60 mod 4 = 0
z mod 4 = 0 → z = 0, 4, 8
Sums:
- 6 + 8 = 14
- 6 + 4 = 10
- 6 + 0 = 6

Step 14: Try y=5:
50 mod 4 = 2
z mod 4 = 2 → z = 2, 6
Sums:
- 5 + 6 = 11
- 5 + 2 = 7

Step 15: Try y=4:
40 mod 4 = 0
z mod 4 = 0 → z = 0, 4, 8
Sums:
- 4 + 8 = 12
- 4 + 4 = 8
- 4 + 0 = 4

Step 16: Try y=3:
30 mod 4 = 2
z mod 4 = 2 → z = 2, 6
Sums:
- 3 + 6 = 9
- 3 + 2 = 5

Step 17: Try y=2:
20 mod 4 = 0
z mod 4 = 0 → z = 0, 4, 8
Sums:
- 2 + 8 = 10
- 2 + 4 = 6
- 2 + 0 = 2

Step 18: Try y=1:
10 mod 4 = 2
z mod 4 = 2 → z = 2, 6
Sums:
- 1 + 6 = 7
- 1 + 2 = 3

Step 19: Try y=0:
0 mod 4 = 0
z mod 4 = 0 → z = 0, 4, 8
Sums:
- 0 + 8 = 8
- 0 + 4 = 4
- 0 + 0 = 0

Step 20: The maximum sum found is 16 for y=8 and z=8.

Step 21: Check if 10y + z = 88 is divisible by 4.
88 ÷ 4 = 22, so yes.

Step 22: Check if the number 45880 is divisible by 40.
45880 ÷ 40 = 1147, which is an integer.

Therefore, the maximum possible value of (y + z) is 16.

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