Number Theory

Greatest Common Divisor / Lowest Common Multiple

GCD and LCM - Problem solving


If the lowest common multiple of 4,6,8,9,4,6,8,9, and xx is 504,504, what is the third smallest possible integer x?x?

Note: Third smallest means the third value of xx if the possible values were listed from least to greatest.

What is the largest 3-digit number that is a multiple of 12 and a multiple of 21, but not a multiple of 22?

If the greatest common divisor of 266,308,392,266,308,392, and xx is 7,7, what is the sum of the smallest and largest possible 2-digit positive integers x?x?

AA and BB are positive integers such that gcd(A,B)=9,lcm(A,B)=180,A<B.\begin{array}{c}&\gcd(A,B)= 9, &\mbox{lcm}(A,B) = 180, &A<B. \end{array} How many such AA's are there?

What is the lowest common multiple of the numbers 8, 9, 10, and 12?


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