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a divides b exactly when every prime in a's factorization appears in b's at least as high a power. Combine divisibility conditions with the LCM, never the product (unless coprime).
💡 Divisible by 6 and by 4 means divisible by lcm = 12, not 24.
Dividing n by d gives n = dq + r with 0 ≤ r < d. Remainders respect addition and multiplication: reduce first, then combine. If the dividend is smaller than the divisor, the quotient is 0 and the remainder is just the dividend itself.
💡 See 'remainder 3 when divided by 7'? Write n = 7k + 3 and test n = 3 first.
The last digit of a product depends only on the inputs' last digits, and last digits of powers repeat in short cycles (length 1, 2, or 4). Find the cycle, then use the exponent mod the cycle length.
💡 Exponent mod 4 = 0 means the LAST position of the cycle, not the first.
Same-base powers: multiply by adding exponents, divide by subtracting, raise a power to a power by multiplying. Zero exponent gives 1; negative exponents flip into reciprocals.
💡 Different bases? Rewrite everything in primes before touching the exponents.
Powers add only when multiplied. A sum of same-base powers is handled by factoring out the smallest power.
💡 $2^n + 2^n = 2^{n+1}$ — doubling adds one to the exponent, nothing else.
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