Find the divisors

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Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\cdot5^5\cdot7^7.$ Find the number of positive integer divisors of n

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Lifting the Exponent shows that

6f16ed711aa8c557c5fbff8616981c6cddf15804

so thus, 819068968ed60608a91d0c3921d6296964660e9d divides 174fadd07fd54c9afe288e96558c92e0c1da733a. It also shows that

a81a4bd215a735a5621a7b3eaf2f6063dd740010

so thus, a525d21eef8ae26634719caaf9bd7a44343c1e6b divides 174fadd07fd54c9afe288e96558c92e0c1da733a.

Now, multiplying 174fadd07fd54c9afe288e96558c92e0c1da733a by c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d, we see

a3c7d40d695af62b96afdda13f1ab8b66d97ba66

and since dba13ba15483a4fbb04b9a70b0e3f70bc6f4ae9c and 424d91de4f8b3505d4e2fd2720e46f9ffa2531cb then 83924c4be361819c9abc4a442ff9e916c2aafa33 meaning that we have that by LTE, ecbe7df4a79ed19fa18091af546030154c4b5491 divides 174fadd07fd54c9afe288e96558c92e0c1da733a.

Since 819068968ed60608a91d0c3921d6296964660e9da525d21eef8ae26634719caaf9bd7a44343c1e6b and f1ab5a3da2efeec25a2b5694c324b69f96fe73dd all divide 174fadd07fd54c9afe288e96558c92e0c1da733a, the smallest value of 174fadd07fd54c9afe288e96558c92e0c1da733a working is their LCM, also 56f754771817eec527fa3ad899a668517cfb958b. Thus the number of divisors is e3ffaf7150cbcc3070db684620490e8f5b6d30b3.

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