Let n be the least positive integer such that 168 is a factor of 1134n. If m is the least positive integer such that 1134n is a factor of 168m, then m + n equals
Explanation:
168 = 23 × 21 = 23 × 3 × 7 1134 = 2 × 567 = 2 × 34 × 7
Now (1134)n = 2n × 34n × 7n Since 168 (23 × 3 × 7) completely divides 1134n (2n × 34n × 7n) ⇒ Power of 2 in 1134n ≥ Power of 2 in 168 ⇒ n ≥ 3 Similarly, we can check for power of 3 and power of 7 and we get the least value of n = 3.
Now (168)m = 23m × 3m × 7m Since 1134n (2n × 34n × 7n = 23 × 312 × 73) completely divides 168m (23m × 3m × 7m) ⇒ Power of 3 in 168m ≥ Power of 3 in 1134n ⇒ m ≥ 12 Similarly, we can check for power of 2 and power of 7 and we get the least value of m = 12.
∴ n + m = 3 + 12 = 15
Hence, option (c).
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