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Math Help - Number Theory questions

  1. #1
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    Number Theory questions

    1.If both 112 and 33 are factors of the number a * 43 * 62 * 1311, then what is the smallest possible value of a?
    2.How many different positive integers exist between 106 and 107, the sum of whose digits is equal to 2?
    3.A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?
    4.How many keystrokes are needed to type numbers from 1 to 1000?
    5.How many integral divisors does the number 120 have?



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  2. #2
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    Re: Number Theory questions

    Quote Originally Posted by pgmedicaladmission View Post
    1.If both 112 and 33 are factors of the number a * 43 * 62 * 1311, then what is the smallest possible value of a?
    2.How many different positive integers exist between 106 and 107, the sum of whose digits is equal to 2?
    3.A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?
    4.How many keystrokes are needed to type numbers from 1 to 1000?
    5.How many integral divisors does the number 120 have?



    Regards,
    PG Medical Admission | Medical PG Admission | PG Medical MD MS Admission | MD MS Admission
    Assuming this is not spam, please show your work so we can see where you are having trouble.

    For problem number 1, I'll give a hint:

    $y = a * 4^3 * 6^2 * 13^{11} = a * (2^2)^3 * (2 * 3)^2 * 13^{11} = a * 2^8 * 3^2 * 13^{11} = 3^2a * 2^8 * 13^{11}.$

    $Let\ b = 3^2a \implies y = b * 2^8 * 13^{11}.$

    $y\ evenly\ divisible\ by\ 3^3\ and\ 11^2 \implies b\ evenly\ divisible\ by\ what?$
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