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Quant · DivisibilityMedium
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Divisibility runs on primes

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).

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Divisibility runs on primes
2376=2333112376 = 2^3 3^3 11: divisible by 24=23324 = 2^3\cdot 3, not by 16=2416 = 2^4

💡 Divisible by 6 and by 4 means divisible by lcm = 12, not 24.

A number divisible by both 10 and 12 must be divisible by which of the following?
Quant · RemaindersMedium
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The remainder identity

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.

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The remainder identity
n=dq+r,  0r<dn = dq + r,\; 0 \le r < d

💡 See 'remainder 3 when divided by 7'? Write n = 7k + 3 and test n = 3 first.

When n is divided by 8 the remainder is 6. What is the remainder when 5n is divided by 8?
Quant · Units DigitsEasy
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Units digits cycle

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.

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Units digits cycle
7n7^n ends in 7,9,3,1,7,9,7, 9, 3, 1, 7, 9, \dots (period 4)

💡 Exponent mod 4 = 0 means the LAST position of the cycle, not the first.

What is the units digit of 2182^{18}?
Quant · Exponent LawsEasy
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The five exponent laws

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.

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The five exponent laws
aman=am+n(am)n=amnan=1/ana^m a^n = a^{m+n} \quad (a^m)^n = a^{mn} \quad a^{-n} = 1/a^n

💡 Different bases? Rewrite everything in primes before touching the exponents.

If 4x8=2114^x \cdot 8 = 2^{11}, then x =
Quant · Factoring PowersMedium
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Sums of powers: factor, don't add exponents

Powers add only when multiplied. A sum of same-base powers is handled by factoring out the smallest power.

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Sums of powers: factor, don't add exponents
312+310=310(32+1)=103103^{12} + 3^{10} = 3^{10}(3^2 + 1) = 10 \cdot 3^{10}

💡 $2^n + 2^n = 2^{n+1}$ — doubling adds one to the exponent, nothing else.

5205185^{20} - 5^{18} is equivalent to which of the following?
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