Thus, there are 14 integers among the first 100 positive integers that are congruent to 3 modulo 7. Therefore, the final answer is:

["Understanding Congruence Modulo 7: Finding How Many of the First 100 Positive Integers Are Congruent to 3 (14 Total)", "When exploring number theory, one fascinating concept is congruence modulo ( n ). In this article, we focus on how many integers among the first 100 positive integers satisfy the condition of being congruent to ( 3 \mod 7 ). Specifically, we’ll explore why exactly 14 such integers exist, and what this means for modular arithmetic.", "### What Does It Mean for an Integer to Be Congruent to 3 Modulo 7?", "A number ( x ) is said to be congruent to ( 3 \mod 7 ) if, when divided by 7, it leaves a remainder of 3. Mathematically, this is expressed as:", "[\nx \equiv 3 \pmod{7}\n]", "This means all integers that can be written in the form:", "[\nx = 7k + 3\n]", "for some integer ( k ), satisfy the congruence.", "### How Many Such Integers Lie Between 1 and 100?", "To find how many integers in the range from 1 to 100 are congruent to ( 3 \mod 7 ), we list the numbers in this sequence:", "Start with ( k = 0 ):\n( 7(0) + 3 = 3 )", "Then increment ( k ) by 1 each time:", "- ( k = 1 ) → ( 7(1) + 3 = 10 )\n- ( k = 2 ) → ( 7(2) + 3 = 17 )\n- ( k = 3 ) → ( 7(3) + 3 = 24 )\n- ( k = 4 ) → ( 7(4) + 3 = 31 )\n- ( k = 5 ) → ( 7(5) + 3 = 38 )\n- ( k = 6 ) → ( 7(6) + 3 = 45 )\n- ( k = 7 ) → ( 7(7) + 3 = 52 )\n- ( k = 8 ) → ( 7(8) + 3 = 59 )\n- ( k = 9 ) → ( 7(9) + 3 = 66 )\n- ( k = 10 ) → ( 7(10) + 3 = 73 )\n- ( k = 11 ) → ( 7(11) + 3 = 80 )\n- ( k = 12 ) → ( 7(12) + 3 = 87 )\n- ( k = 13 ) → ( 7(13) + 3 = 94 )\n- ( k = 14 ) → ( 7(14) + 3 = 101 ) → but 101 exceeds 100, so it’s out of range", "Counting the valid values:\nFrom ( k = 0 ) to ( k = 13 ), we have 14 integers:", "[\n3,\ 10,\ 17,\ 24,\ 31,\ 38,\ 45,\ 52,\ 59,\ 66,\ 73,\ 80,\ 87,\ 94\n]", "Thus, there are exactly 14 integers between 1 and 100 that are congruent to 3 modulo 7.", "### Why Are Exactly 14 Numbers Satisfying the Condition?", "Each number in the sequence increases by 7 from the previous one (since adding 7 preserves the remainder modulo 7). Starting from 3 (which is ( \equiv 3 \mod 7 )), we keep adding 7 until we exceed 100.", "The largest such number under 100 is ( 94 = 7 \ imes 13 + 3 ). The smallest is 3 (( 7 \ imes 0 + 3 )), so the values of ( k ) run from 0 to 13 — a total of 14 integers.", "This confirms:\nThus, there are 14 integers among the first 100 positive integers congruent to 3 modulo 7.", "### The Final Answer:", "Therefore, the final answer is: 14", "---", "This modular insight reveals how number patterns emerge predictably under division rules. Understanding congruences helps in areas ranging from cryptography and computer science to scheduling and pattern analysis — showcasing the power and elegance of number theory in everyday applications."]