Number Theory

Maze wrote:If both 11^2 and 3^3 are factors of the number a * 4^3 * 6^2 * 13^11, then what is the smallest possible value of a?
A. 121
B. 3267
C. 363
D. 33
E. None of the above

IMO C

11^2 is a factor of a * 4^3 * 6^2 * 13^11.

The number does not have a power or multiple of 11 as its factor.

Hence, "a" should include 11^2

3^3 is a factor of the given number. [6^2 = (3x2)^2 ]

Therefore, if 3^3 has to be a factor of the given number a * 4^3 * 6^2 * 13^11, then we will need at least another 3.

Therefore, "a" should be at least 11^2 * 3 = 363 if the given number has to have 11^2 and 3^3 as its factors.

Maze wrote:If both 11^2 and 3^3 are factors of the number a * 4^3 * 6^2 * 13^11, then what is the smallest possible value of a?
A. 121
B. 3267
C. 363
D. 33
E. None of the above

least value of a = 11^2*3 [3^2 is already a factor]

a=363

Maze wrote:If both 11^2 and 3^3 are factors of the number a * 4^3 * 6^2 * 13^11, then what is the smallest possible value of a?
A. 121
B. 3267
C. 363
D. 33
E. None of the above

a * 4^3 * 6^2 * 13^11 = a * 2^6 * (3X2)^2 * 13^11 = a* 2^8 * 3^2*13^11
since 11^2 and 3^3 are factor of above exp
so a= 3x121=363 Ans C