Overview

A problem where you convert given “hour” and “minute” pairs into correct date/time format (days later, hours, minutes).

Approach

This is a fundamental problem that can be solved using unit conversion and quotient/remainder calculations.

First, since the minute value \(M\) can be input as \(60\) or greater, we need to properly convert this into “carry-over to hours.”
For example, \(80\) minutes is \(1\) hour and \(20\) minutes, so we add \(80 \div 60 = 1\) to the hours and use the remainder \(80 \bmod 60 = 20\) as the new minute value.

Next, if the updated hour value \(H'\) is \(24\) or greater, this means there is a “carry-over to days.”
For example, \(25\) hours is \(1\) day and \(1\) hour, so \(25 \div 24 = 1\) day with a remainder of \(25 \bmod 24 = 1\) hour.

In this way, by sequentially performing quotient (//) and remainder (%) calculations, we can obtain the normalized days, hours, and minutes.

Note that according to the constraints, \(H, M\) can be up to \(10^9\), so a naive simulation (e.g., incrementing by one minute in a loop) will result in TLE. A mathematical approach that computes the answer directly is required.

Algorithm

We normalize each time data using the following steps:

1. Carry-over from minutes

   • Set the new hours \(H'\) to \(H + \left\lfloor \frac{M}{60} \right\rfloor\).
   • Set the new minutes \(m\) to \(M \bmod 60\).

2. Carry-over from hours

   • Set the number of days \(D\) to \(\left\lfloor \frac{H'}{24} \right\rfloor\).
   • Set the normalized hours \(h\) to \(H' \bmod 24\).

Finally, output \(D\), \(h\), \(m\).

Concrete Example

For input 25 80: - Carry-over from minutes: \(H' = 25 + \left\lfloor \frac{80}{60} \right\rfloor = 25 + 1 = 26\), \(m = 80 \bmod 60 = 20\) - Carry-over from hours: \(D = \left\lfloor \frac{26}{24} \right\rfloor = 1\), \(h = 26 \bmod 24 = 2\)

Therefore the output is 1 2 20

Complexity

• Time complexity: \(O(N)\)
• Space complexity: \(O(1)\)

Since each data point can be processed in constant time, the overall solution runs in linear time.

Implementation Notes

• Quotient and remainder calculations are sufficiently fast with integer arithmetic.

• Since the input range is very large (up to \(10^9\)), do not simulate using loops or similar approaches.

• Organizing variable names like \(H', h, m, D\) and making their meanings clear will improve code readability.

  Source Code

N = int(input())
for _ in range(N):
    H, M = map(int, input().split())
    # 分の繰り上げ
    H += M // 60
    m = M % 60
    # 時の繰り上げ
    D = H // 24
    h = H % 24
    print(D, h, m)

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