Number Theory Checkpoint

Check Euclid’s invariant, pre-multiplication overflow, shared result semantics, modular boundaries, and exclusive prime counting. Select one option per multiple-choice question.

Get every question correct to pass this checkpoint. You can review feedback, revise your answers, and submit again.

1. For a=91 and b=35, Euclid replaces (91,35) with (35,21). Why is the GCD unchanged? Select one answer.

Select one answer.

Because the sum of the pair stays constant Because every common divisor of either pair divides both values of the other pair Because both numbers always decrease by the same amount Because the remainder is always prime

2. A positive current LCM is L, the next value is v, and q=L/gcd(L,v). With MAX=2^63-1, which step safely detects an unrepresentable next LCM? Select one answer.

Select one answer.

Multiply q*v, then check whether it is negative Check whether L exceeds v If q > MAX/v using integer division, record overflow before multiplying Divide the wrapped product by v to recover it

3. The mathematical LCM of an allowed array is above 2^63-1. Python can represent it exactly. What must the Python exercise return in its LCM position? Select one answer.

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The exact unbounded integer 0 The value modulo 2^63 -1

4. An array prefix has GCD 2 and its LCM has already overflowed. The only remaining value is 3. What should the final pair be? Select one answer.

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[1,-1] [2,-1] [-1,-1] [1,3]

5. What is 7^4 mod 13? Enter one integer.

6. A modular-power function starts result at 1 and skips the loop when exp=0. What initialization also handles mod=1 correctly? Select one answer.

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result=base result=1 % mod result=exp % mod result=mod

7. How many primes are strictly less than 17? Enter one integer.

8. When a sieve reaches prime p=5, why may marking begin at 25 instead of 10? Select one answer.

Select one answer.

Numbers below 25 are all prime The sieve is inclusive of its upper bound Every smaller composite multiple of 5 has already been marked through a smaller prime factor Multiples of 5 never need marking

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