A complete English student edition of Test 16 from this exercise. Record a final response for every problem and use the optional work spaces to show your remainder equations, candidate lists, or cycle reasoning.
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Division conventions: the divisor is positive; the quotient and remainder are nonnegative whole numbers, with 0 ≤ r < d. A three-digit integer is from 100 through 999. “Positive” excludes zero.
Questions 1–12 · 10 points each
A class with fewer than 60 students bought 310 exercise books. Every student receives as many whole books as possible, with equal numbers for everyone and 37 books left over. How many students are in the class?
When 474 is divided by a two-digit number, the remainder is 6. List all two-digit divisors that satisfy this condition.
A nonnegative whole number a leaves remainder 3 when divided by 7. A nonnegative whole number b leaves remainder 4 when divided by 7. What remainder does a+b leave when divided by 7?
A nonnegative whole number leaves remainder 3 when divided by 7. What remainder does three times that number leave when divided by 7?
When one nonnegative whole number is divided by a positive whole number, the quotient is 4 and the remainder is 8. The dividend, divisor, quotient, and remainder add to 415. What is the dividend?
Find the least positive whole number that leaves remainder 1 when divided by 3, remainder 2 when divided by 4, remainder 3 when divided by 5, and remainder 4 when divided by 6.
Observe the sequence:
Beginning with the third term, each term is the sum of the two preceding terms. What remainder does the 2,014th term leave when divided by 3?
Two different three-digit numbers leave the same remainder when divided by 13.
What is the greatest possible sum of the two numbers? What is the greatest possible difference (larger minus smaller) between them? Find the two maxima separately; they do not have to come from the same pair.
When one nonnegative whole number is divided by a positive whole number, the quotient is 17 and the remainder is 13. The dividend, divisor, quotient, and remainder add to 2,113. What is the dividend?
A positive integer has the following two properties:
What is the positive integer?
A cyclist trains between places A and B, which are 950 km apart. Starting from A, the cyclist rests every 90 km on the outward trip. After reaching B, the cyclist rests for one day and returns along the same route, resting every 100 km. Only rest points along the route are counted; exclude the endpoints A and B. Exactly one return-trip rest point is also an outward-trip rest point.
How far is this common rest point from A?
Choose any two-digit number. Repeat that same two-digit block three more times, producing an eight-digit number made from four identical blocks.
Divide the eight-digit number by the original two-digit number. Then divide the resulting quotient by 9. What remainder is obtained?