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24

Part of 2023 AMC 12/AHSME

Problems(2)

suspiciously sequenced sub-sets

Source: 2023 AMC12A #24

11/9/2023
Let KK be the number of sequences A1A_1, A2A_2, …\dots, AnA_n such that nn is a positive integer less than or equal to 1010, each AiA_i is a subset of {1,2,3,…,10}\{1, 2, 3, \dots, 10\}, and Ai−1A_{i-1} is a subset of AiA_i for each ii between 22 and nn, inclusive. For example, {}\{\}, {5,7}\{5, 7\}, {2,5,7}\{2, 5, 7\}, {2,5,7}\{2, 5, 7\}, {2,5,6,7,9}\{2, 5, 6, 7, 9\} is one such sequence, with n=5n = 5. What is the remainder when KK is divided by 1010?
<spanclass=′latex−bold′>(A)</span>1<spanclass=′latex−bold′>(B)</span>3<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>7<spanclass=′latex−bold′>(E)</span>9<span class='latex-bold'>(A) </span> 1 \qquad <span class='latex-bold'>(B) </span> 3 \qquad <span class='latex-bold'>(C) </span> 5 \qquad <span class='latex-bold'>(D) </span> 7 \qquad <span class='latex-bold'>(E) </span> 9
AMCAMC 122023 AMC2023 AMC 12ASequencesremainderSets
Misplaced much?

Source: 2023 AMC 12B p24

11/15/2023
Integers a,b,c,da, b, c, d satisfy the following: abcd=26⋅39⋅57abcd=2^6\cdot 3^9\cdot 5^7 lcm(a,b)=23⋅32⋅53\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3 lcm(a,c)=23⋅33⋅53\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3 lcm(a,d)=23⋅33⋅53\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3 lcm(b,c)=21⋅33⋅52\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2 lcm(b,d)=22⋅33⋅52\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2 lcm(c,d)=22⋅33⋅52\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2
Find gcd(a,b,c,d)\text{gcd}(a,b,c,d) <spanclass=′latex−bold′>(A)</span> 30<spanclass=′latex−bold′>(B)</span> 45<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 15<spanclass=′latex−bold′>(E)</span> 6<span class='latex-bold'>(A)</span>~30\qquad<span class='latex-bold'>(B)</span>~45\qquad<span class='latex-bold'>(C)</span>~3\qquad<span class='latex-bold'>(D)</span>~15\qquad<span class='latex-bold'>(E)</span>~6
2023 AMC 12BAMCAMC 12number theoryleast common multiplegreatest common divisor2023 AMC