A number when divided by 7, 11 and 13 (the prime factor of 1001) successively leave the remainders 6, 10 and 12 respectively. Find the remainder if the number is divided by 1001.

Let the unknown number be N. The number N divided by 7 gives a quotient a and a remainder 6, so N = 7a + 6. Substituting this into the second condition gives 7a + 6 = 11b + 10, and applying the third condition yields the combined remainder formula N = 7 * 11 * 13 * q + 12 * 77 + 10 * 7 + 6. Evaluating the constant terms gives 924 + 70 + 6 = 1000, which is less than 1001, so the remainder is 1000.

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