Find a two-digit positive integer that is three less than a multiple of 7 and three less than a multiple of 8.
["Explore All Two-Digit Positive Integers That Are Three Less Than a Multiple of Both 7 and 8", "If you’re searching for a two-digit positive integer that is three less than a multiple of both 7 and 8, you’ve come to the right place. This unique mathematical condition leads to a precise and elegant solution.", "---", "### What Does It Mean to Be “Three Less Than a Multiple of 7 and 8”?", "A number that is three less than a multiple of both 7 and 8 satisfies:", "[\nn \equiv -3 \pmod{7} \quad \ ext{and} \quad n \equiv -3 \pmod{8}\n]", "This is equivalent to:", "[\nn + 3 \equiv 0 \pmod{7} \quad \ ext{and} \quad n + 3 \equiv 0 \pmod{8}\n]", "So, ( n + 3 ) must be a common multiple of 7 and 8. Since 7 and 8 are coprime, their least common multiple is:", "[\n\ ext{lcm}(7, 8) = 7 \ imes 8 = 56\n]", "Thus,", "[\nn + 3 = 56k \quad \ ext{for some integer } k\n]", "Then:", "[\nn = 56k - 3\n]", "We now seek all two-digit integers ( n ), so we solve:", "[\n10 \leq 56k - 3 \leq 99\n]", "Add 3 to all parts:", "[\n13 \leq 56k \leq 102\n]", "Divide by 56:", "[\n\frac{13}{56} \approx 0.23 \leq k \leq \frac{102}{56} \approx 1.82\n]", "So, the only integer ( k ) in this range is ( k = 1 ), giving:", "[\nn = 56(1) - 3 = 53\n]", "---", "### Verification", "Check:\n- ( 53 + 3 = 56 )\n- ( 56 \div 7 = 8 ) ✔\n- ( 56 \div 8 = 7 ) ✔", "Thus, 53 is indeed three less than a multiple of both 7 and 8.", "---", "### Final Answer", "The only two-digit positive integer that is three less than a multiple of both 7 and 8 is 53.", "---", "### Why This Problem Matters", "This search highlights how modular arithmetic and least common multiples simplify solving for numbers satisfying congruence-based conditions. Whether for math enthusiasts, educators, or students, such problems reinforce core concepts in number theory and problem-solving skills.", "For more like this, explore how different offsets with respect to multiples of small integers reveal patterns across the number line.", "---", "Keywords: Find two-digit number, multiple of 7 and 8, three less than multiple, modular arithmetic, common multiple, math problem solution, academic exercises, number theory clues."]
