Number Theory

Greatest Common Divisor / Lowest Common Multiple

Greatest Common Divisor / Lowest Common Multiple: Level 3 Challenges

           

If xx and yy are integers, what is the smallest possible positive value of 30x+18y?30x+18y?

2,4\huge \color{#D61F06}{2}, \color{#20A900}{-4}

Find the lowest common multiple of the two numbers above.

0.3ˉ,0.5ˉ\LARGE \color{#3D99F6}{0.\bar{3}}, \color{#D61F06}{0.\bar{5}}

What is the lowest common multiple of the two numbers given above?

Note that 0.ABC=0.ABCABCABC0. \overline{ABC} = 0.ABCABCABC\ldots .

How many positive integers aa are there such that 2027 divided by aa leaves a remainder of 7?

Evaluate gcd(19!+19,20!+19) \gcd ( 19 ! + 19, 20! + 19 ) .

Details and assumptions

The number n! n!, read as n factorial, is equal to the product of all positive integers less than or equal to nn. For example, 7!=7×6×5×4×3×2×1 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1.

How many ordered pairs of positive integers (m,n) (m, n) satisfy

gcd(m3,n2)=2232 and lcm(m2,n3)=243456? \gcd (m^3, n^2) = 2^2 \cdot 3^2 \text{ and } \text{lcm}(m^2, n^3) = 2^4 \cdot 3^4 \cdot 5^6?

This problem is shared by Muhammad A.

Details and assumptions

gcd(a,b) \gcd(a, b) and lcm(a,b) \text{lcm}(a, b) denote the greatest common divisor and least common multiple of a a and b b , respectively.

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