Number Theory D3-2
6
1. What is the 453rd digit to the right of the decimal point in the decimal expansion of 13 ?
2. The decimal representation of a fraction ends in 0.3. When the decimal is changed to a
common fraction and reduced to lowest terms, what is the denominator of the fraction?
3. When the two-digit integer M M , with equal digits, is multiplied by the one-digit integer M ,
the result is the three-digit integer N P M . What is the greatest possible value of N P M ?
4. Karla writes down six different prime numbers in increasing order. She notices that the
product of the first three prime numbers she has written is equal to the sum of the last three prime numbers
she has written. What is the least possible value of the last prime number Karla wrote?
5. What is the units digit of 19981998 ?
6. If the three-digit number 2d2 is divisible by 7, what is d?
7. How many positive integers are factors of 2175 ?
8. How many positive even three-digit integers are divisible by 3?
9. Select two different numbers from the set {4, 9, 12, 16, 18}. Of all possible pairs, what is the
least possible value for the sum of the greatest common factor and least common multiple of the two
numbers?
10. The product of the digits of a three-digit number is 63. What is the greatest possible value of
the number?
11. When two different numbers are divided by 7, remainders of 2 and 3, respectively, are left.
What is the greatest possible three-digit product of these two numbers?
12. What is the tens digit of the product of the first six prime numbers?
13. What is the base five product of the numbers 1215 and 115 ?
2240
14. If nk is the kth digit to the right of the decimal point in the decimal representation of 1111 ,
what is the value of 1000n16 + 100n13 + 10n10 + n7 ?
15. The number 839 can be written as 19q + r where q and r are positive integers. What is the
greatest possible value of q − r?
16. What is the sum of the tens digit and the units digit in the decimal representation of 92004 ?
17. The product of a set of distinct positive integers is 84. What is the least possible sum of these
integers?
18. Given that 3n divides 15!, what is the greatest possible integral value of n?
19. When an integer is divided by 15, the remainder is 7. Find the sum of the remainders when
the same integer is divided by 3 and by 5.
20. If a three-digit number of the form 1D1 is divided by D, the quotient is a two digit number of
the form 2D remainder 5. What is the value of D?
21. What is the ones digit of 735 when written as an integer?
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