1) In the following sequence of numbers, each number has one more 1 than the preceding number: 1, 11, 111, 1111, 11111, ... What is the tens digit of the sum of the first 30 numbers of the sequence?
2) When asked how many gold coins he had, the collector said:
If I arrange them in stacks of five, none are left over.
If I arrange them in stacks of six, none are left over.
If I arrange them in stacks of seven, one is left over.
What is the least number of coins he could have?
3) In the subtraction problem below, each letter represents a digit, and different letters represent different digits. What digit does C represent?
A B A
- C A
1) The ones column of the 30 numbers contains 30 ones making a sum of 30. Thus, the ones digit of the sum is 0, carry 3. The tens column contains 29 ones. Its sum is 29 plus the 3 from the "carry," making 32. Therefore the tens digit of the sum is 2.
2) The number of coins must be a multiple of 30. The multiples of 30 are 30, 60, 90, 120, 150, and so on. The smallest of these multiples that leaves a remainder of 1 when divided by 7 is 120.
3) It is clear that B = 0 and A = 1. Substitute those numbers for letters as shown below. Therefore C is 9.
1 0 1
- C 1