Let n be the least positive integer such that 168 is a factor of 1134^n. If m is the least positive integer such that 1134^n is a factor of 168^m, then m+n equals

Question

UPSC Quizroom · Accepted Answer

Factorize the numbers:

168 = 2^3 * 3^1 * 7^1

1134 = 2^1 * 3^4 * 7^1

Let n be minimal with 168 | 1134^n. For each prime p we need n * v_p(1134) ≥ v_p(168):

for 2: n*1 ≥ 3 → n ≥ 3

for 3: n*4 ≥ 1 → n ≥ 1

for 7: n*1 ≥ 1 → n ≥ 1
So n = 3.

Now find minimal m with 1134^n | 168^m. Exponents in 1134^n (with n=3) are:

2: 3, 3: 12, 7: 3.
Require m * v_p(168) ≥ those:

for 2: m*3 ≥ 3 → m ≥ 1

for 3: m*1 ≥ 12 → m ≥ 12

for 7: m*1 ≥ 3 → m ≥ 3
So m = 12.

Therefore m + n = 12 + 3 = 15.

🎯 UPSC Quiz Room

Let n be the least positive integer such that 168 is a factor of 1134^n. If m is the least positive integer such that 1134^n is a factor of 168^m, then m+n equals

Options

Recommended articles

Let n be the least positive integer such that 168 is a factor of 1134^n. If m is the least positive integer such that 1134^n is a factor of 168^m, then m+n equals

Options

Recommended articles

📱 Follow Us