#### 78.4Question: What is the smallest positive integer that leaves a remainder of 3 when divided by 5 and a remainder of 4 when divided by 7? - Treasure Valley Movers
What is the smallest positive integer that leaves a remainder of 3 when divided by 5 and a remainder of 4 when divided by 7?
What is the smallest positive integer that leaves a remainder of 3 when divided by 5 and a remainder of 4 when divided by 7?
Curiosity about patterned numbers and problem-solving has sparked interest in modular math puzzles—especially since this classic question shows up in everyday situations involving timing, scheduling, and trends. The search for the smallest positive integer satisfying both a remainder of 3 mod 5 and 4 mod 7 reveals a simple but elegant number theory concept used across finance, planning, and digital tools. Many U.S. users are drawn to these kinds of logic challenges as they connect to apps, routines, and system checks they rely on daily. This query reflects how intelligent pattern recognition supports better decisions, whether managing personal schedules or building reliable systems.
The problem asks for the smallest positive integer that, when divided by 5, leaves a remainder of 3, and when divided by 7, leaves a remainder of 4. This is a modular equation commonly solved using the method of successive substitutions or the least common solution principle. Mathematically, the two conditions can be written as:
x ≡ 3 (mod 5)
x ≡ 4 (mod 7)
Understanding the Context
To solve, start by listing numbers that satisfy the first condition: 3, 8, 13, 18, 23, 28, 33, 38… Next, identify which of these leave a remainder of 4 when divided by 7. Checking each:
3 ÷ 7 → remainder 3
8 ÷ 7 → remainder 1
13 ÷ 7 → remainder 6
18 ÷ 7 → remainder 4 ✔ — this fits both conditions.
So, 18 is the smallest
