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Home » \mathbfM = \beginpmatrix a & b \\ c & d \endpmatrix - ToelettAPP

\mathbfM = \beginpmatrix a & b \\ c & d \endpmatrix - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Check for $3 \cdot 9 = 9$: yes.

📌 But what about $3 \cdot 5 = 15$? We saw $120$ in previous problem, but here product $1 \cdot 3 \cdot 5 \cdot 7 = 105$, divisible by $15$, but $3 \cdot 5 = 15$ — yes. But $3 \cdot 3 = 9$: $105$ not divisible by $9$. So $9$ does not always divide $P$.

📌 So far, guaranteed divisors: $3$. But is $3 \cdot 3 = 9$ always? No.

📌 But consider modulo $9$: the multiple of 3 in the four may be $3 \mod 9$ or $6 \mod 9$, but not $0$ or $9$. Only when $3k$ where $k$ divisible by 3. So not guaranteed.

📌 Now consider **product modulo small numbers**. Try small cases:

📌 $1 \cdot 3 \cdot 5 \cdot 7 = 105$

📌 $3 \cdot 5 \cdot 7 \cdot 9 = 945$

📌 $5 \cdot 7 \cdot 9 \cdot 11 = 3465$

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