A complete English student edition of Test 19. Work through remainder systems, GCD and LCM structures, equal sharing and cutting, cube construction, reverse GCD–LCM conditions, and complete counting arguments.
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Questions 1–10 · 6 points each · 60 points
A grade has from 200 through 300 students. When the students are arranged:
How many students are in the grade?
Let a=b−1, where a and b are natural numbers and a≠0. Find their greatest common divisor and least common multiple.
Two natural numbers have sum 50 and greatest common divisor 5. Determine every possible value of the larger number minus the smaller number.
Every person at a gathering shares one rice bowl, one vegetable bowl, and one soup bowl. For each kind of food, everyone is placed in groups of exactly the stated size, with no incomplete group. The three kinds use separate bowls:
A total of 65 bowls are used. How many people attend?
There are 252 red balls, 396 blue balls, and 468 yellow balls. Pack all of them into n nonempty bags so that every bag contains all three colors and every bag has the same number of red balls, the same number of blue balls, and the same number of yellow balls.
What is the greatest possible value of n?
Identical rectangular wooden blocks measure:
Use whole blocks to build one large solid cube. Arrange the blocks in straight rows and layers, all facing the same way: all 5 cm edges are parallel to one cube edge, all 4 cm edges to another, and all 3 cm edges to the third. Leave no gaps and do not cut any blocks. What is the least number of blocks required?
Arrangement clarification: this exercise uses a regular array so you can apply the LCM method.
What is the greatest positive integer that divides 30, 60, and 75 exactly?
Three wires have lengths 120 cm, 180 cm, and 300 cm. Cut every wire into equal sections of the greatest possible length, with no wire left over.
Find the section length and the total number of sections.
A natural number greater than 1 is used as the divisor of 326, 258, and 207. All three divisions leave the same remainder. Find the divisor.
From 60 through 100 students are taking part in a gymnastics performance. They can be divided evenly into groups of 8, and they can also be divided evenly into groups of 12.
What is the least possible number of students?
Questions 11–13 · 20 points each · 60 points
Use each of the digits below exactly once to form two three-digit numbers:
Make the greatest common divisor of the two three-digit numbers and 540 as large as possible. Give the two numbers and the maximum common divisor. Either order for the two numbers is acceptable.
The shared divisor must divide 540. List its divisors, then test the larger ones first. Can two three-digit multiples use exactly the six given digits? To prove your maximum, explain why every larger divisor fails.
Positive integers a, b, and c satisfy:
How many triples (a,b,c) satisfy all three conditions?
Each of a, b and c must divide 90. Also, a and b must be multiples of 10. List the ordered pairs (a,b) whose GCD is exactly 10, then test which divisors c satisfy both LCM conditions. Use a table to avoid missing or counting a triple twice.
How many positive integers n satisfy: