So the only way is to find smallest m such that 45m is divisible by 18 and 45m not divisible by 90.

["Title: Finding the Smallest Positive Integer m Such That 45m Is Divisible by 18 but Not by 90", "---", "Introduction", "In mathematics, especially number theory, finding the smallest integer that satisfies specific divisibility conditions is a fascinating challenge. In this article, we explore the precise condition: find the smallest value of m such that 45m is divisible by 18 but NOT divisible by 90. This problem combines concepts of divisibility, prime factorization, least common multiples (LCM), and modular reasoning.", "---", "### Understanding the Problem", "We are given:", "> Find the smallest positive integer ( m ) such that \n\n( 18 \mid 45m ) (i.e., 45m is divisible by 18),\nbut ( 90 <br/>\nmid 45m ) (i.e., 45m is not divisible by 90).", "Let’s unpack this logically and mathematically.", "---", "### Step 1: Analyze Divisibility by 18 and 90", "First, recall the prime factorizations:", "- ( 18 = 2 \ imes 3^2 )\n- ( 90 = 2 \ imes 3^2 \ imes 5 )", "So the condition translates to:", "- ( 45m ) must include at least one factor of 2, two factors of 3, and one factor of 5 for divisibility by 18.\n\nBut crucially, ( 45m ) must lack the full factor of 5 required for divisibility by 90 — or more precisely, the total number of 2s, 3s, or 5s in ( 45m ) must fall short of what 90 demands.", "But since ( 45 = 3^2 \ imes 5 ), the prime factors of ( 45m ) are:", "- Existing from 45: ( 3^2 \ imes 5 )\n- Plus the factors of ( m )", "So:", "- Number of 2s in ( 45m ) = number of 2s in ( m )\n- Number of 3s in ( 45m ) = ( 2 + \ ext{powers of 3 in } m )\n- Number of 5s in ( 45m ) = ( 1 + \ ext{powers of 5 in } m )", "---", "### Step 2: Apply Divisibility by 18", "For ( 45m ) to be divisible by 18, the exponents of primes in ( 45m ) must meet or exceed those in 18:", "- ( v_2(45m) \geq 1 ) → since ( 45 ) has no 2s, ( m ) must contribute at least one factor of 2\n- ( v_3(45m) \geq 2 ) → since 45 contributes ( 3^2 ), ( m ) may contribute extra 3s, but even 2 suffices\n- ( v_5(45m) \geq 1 ) → already satisfied by 45, so extra is fine", "Therefore, the minimal requirement is that ( m ) must include at least one factor of 2. So:", "[\nm \ ext{ must be divisible by } 2\n]", "But we cannot stop here — we also require that ( 45m ) is not divisible by 90.", "---", "### Step 3: Apply Non-Divisibility by 90", "We now enforce that ( 45m ) is not divisible by 90.", "Recall: ( 90 = 2 \ imes 3^2 \ imes 5 )", "Currently, ( 45m ) already has ( 3^2 ) and ( 5 ) from the 45. So, for ( 90 \mid 45m ), the only missing real constraint is the power of 2.", "Wait — but 90 requires exactly the primes of 2, 3², and 5. Since ( 45m ) already has ( 3^2 \ imes 5 ), the only additional condition that could block divisibility by 90 is an insufficient power of 2 — but actually, worse: if ( 45m ) lacks at least one prime factor of 90, it fails.", "But all are satisfied from 45 — so:", "- If ( 45m ) has no factor of 2, then it cannot be divisible by 90, even if it has ( 3^2 \ imes 5 )", "Thus, to not be divisible by 90, the critical missing requirement is that ( m ) has no factor of 2 → i.e., ( v_2(m) = 0 )", "But wait — if ( m ) includes a factor of 2, then ( 45m ) has ( v_2 \geq 1 ), so divisible by 2, and with ( 3^2 ) and ( 5 ), it becomes divisible by ( 2 \ imes 3^2 \ imes 5 = 90 ), which violates the second condition.", "Therefore:", "> If ( m ) is even, then ( 45m ) is divisible by 2, and since it contains ( 3^2 ) and ( 5 ), it is divisible by 90 → violates the "not divisible by 90" condition", "Hence, to avoid divisibility by 90, ( m ) must not be divisible by 2.", "---", "### Step 4: Conscile Both Conditions", "From divisibility by 18:\n( m ) must be divisible by 2", "From non-divisibility by 90:\n( m ) must not be divisible by 2", "This seems contradictory — but only if both are strictly enforced.", "So clearly, both cannot be true simultaneously — unless we reevaluate.", "Wait: this suggests our earlier logic needs refinement.", "Let’s recheck:", "- To be divisible by 18: needs ( 2^1, 3^2, 5^1 )\n ( 45 = 3^2 \cdot 5 ), so ( m ) must supply at least ( 2^1 )", "- To not be divisible by 90: cannot have enough of any prime factoring together — especially, to avoid being divisible by 90, the full factorization must fail to meet at least one required exponent.", "But 90 = ( 2 \ imes 3^2 \ imes 5 ). The problem is that even though ( 45m ) has ( 3^2 ) and ( 5 ), it might still be divisible by 90 if it has a factor of 2 — which it gains if ( m ) is even.", "But if ( m ) is even → ( v_2(45m) \geq 1 ), and ( v_3 = 2 ), ( v_5 = 1 ) → divisible by 90 → violates condition.", "Hence, to avoid divisibility by 90, ( m ) must not have a factor of 2 — so ( v_2(m) = 0 )", "But then ( v_2(45m) = 1 ), so divisible by 2 → but not by 5? Wait: ( 45 ) already has 5, so ( 45m ) has factor 5 → so divisible by 2, 3², 5 → divisible by 90 → again contradiction.", "Hence, if ( m ) is not even, then no factor of 2 → ( v_2(45m) = 1 ), and ( 45m ) divisible by ( 2 \ imes 3^2 \ imes 5 = 90 ) → fails "not divisible by 90" condition.", "If ( m ) is even, then ( v_2 \geq 1 ), so ( v_2(45m) \geq 1 ), and ( v_3 \geq 2 ), ( v_5 = 1 ) → divisible by 90 → fails divisibility-by-18 condition.", "So both conditions cannot be satisfied?", "Wait — this suggests no solution exists?", "But that contradicts the idea of the problem — so where is the mistake?", "Ah! Here’s the key: divisibility by 90 requires all three prime powers, but if 45m already has all three components, then to not be divisible by 90, one of the required exponents must be missing.", "But 45 already supplies ( 3^2 ) and ( 5 ). So only missing necessarily is ( v_2 ).", "Thus:", "- If ( m ) is even → ( v_2 \geq 1 ) → ( 45m ) divisible by 2, 3², 5 → divisible by 90 → fails\n- If ( m ) is odd → ( v_2 = 0 ) → ( 45m ) not divisible by 2 → cannot be divisible by 90 → satisfies non-divisibility\n- But ( v_2 < 1 ) → fails divisibility by 18", "So no value of m satisfies both?", "But that can't be.", "Wait — perhaps the error is in assuming "not divisible by 90" only fails due to lack of 2.", "But 90 requires all three — so if even one** of ( 2^1, 3^2, 5^1"]