1
Count perfect squares through 300
Not answeredAmong the positive integers from 1 through 300, how many are perfect squares?
squares
Lesson 20.7 — Chapter 20 Exercise
A complete English student edition of Test 20. Work through square counts, least multipliers, factorial and digit-pattern questions, square endings, shifts to nearby squares, sums of squares, and interval-counting arguments.
Exercise directions
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Tips for this exercise
Perfect-square conventionsA perfect square is a number of the form r² = r × r, where r is a nonnegative whole number. This includes 0² = 0. “Three-digit” means 100–999 and “four-digit” means 1000–9999. Unless a question says otherwise, variables are positive whole numbers.
Read the notation carefullyQuestion 3 means n! + 3 with n > 1. In Question 8, ab and ba are two-digit numerals, not products.
Section I · Fill-in questions
Questions 1–10 · 6 points each · 60 points
2
Supply the least square-completing factor
Not answeredLet x be a positive integer. If 32x is a perfect square, what is the least possible value of x?
Optional prime-exponent work
3
A factorial expression that becomes a square
Not answeredFor a whole number n>1, suppose:
n! + 3 is a perfect square,
where n! = 1 × 2 × 3 × ··· × n.
where n! = 1 × 2 × 3 × ··· × n.
What is the least possible value of n?
Typesetting clarificationThe worked example line is read as n!+3, together with the condition n>1. It is not n!+3(n−1).
Optional candidate check
4
Match a three-digit square to a digit equation
Not answeredFind every three-digit perfect square for which:
hundreds digit + ones digit = tens digit.
List every qualifying square, separated by commas, in any order.
Optional square list or search
5
Impossible ones digits of squares
Not answeredSome decimal digits can never appear as the ones digit of a perfect square. What is the sum of all those impossible ones digits?
Optional units-digit table
6
Three-digit squares with tens digit 7
Not answeredA three-digit perfect square has 7 as its tens digit. Find all such perfect squares.
Optional root search
7
Add a two-digit prime to reach a square
Not answeredAdding a two-digit prime number to 2002 produces a perfect square. What is the prime?
Optional nearby-square work
8
A two-digit number and its reversal
Not answeredLet ab and ba be two-digit numerals, where a<b. If:
ab + ba is a perfect square,
how many possible two-digit numerals ab are there?
Numeral conventionab means 10a+b, not a×b. Because both written forms are two-digit numerals, neither leading digit is zero.
numbers
Optional digit-equation work
9
Maximize a square sum of three distinct numbers
Not answeredThree different two-digit positive integers have a sum that is a perfect square. What is the greatest possible value of their sum?
Optional upper bound and construction
10
The least square centered in three consecutive integers
Not answeredThree consecutive positive integers are all four-digit numbers. The middle integer is a perfect square, and the sum of the three integers is divisible by 15.
What is the least possible value of the middle integer?
Optional divisibility and square-root work
Section II · Extended-response questions
Questions 11–14 · 15 points each · 60 points
11
Scale a known sum of squares
Not answeredYou are given:
1² + 2² + 3² + ··· + 25² = 5525.
Find:
3² + 6² + 9² + ··· + 75².
Optional strategy hint
Write each term as the square of 3 times a whole number. What common factor can you take outside the sum? Apply it to the given total.
12
The first square-sum divisible by 5
Not answeredLet n be a positive integer. If:
5 divides 1² + 2² + ··· + n²,
what is the least possible value of n?
Optional strategy hint
Check the partial sums in order, starting with n=1. The first one divisible by 5 gives a candidate; explain why every smaller allowed n fails.
13
Count three-digit shifts from 2010 to a square
Not answeredHow many three-digit positive integers k satisfy:
2010 + k is a perfect square?
Here 100≤k≤999.
Optional strategy hint
Add 2010 to both endpoints of the three-digit range. Find the first and last whole-number roots whose squares fall inside the resulting interval, then count those roots inclusively.
values
14
Count four-digit numbers that become squares after adding 400
Not answeredHow many four-digit positive integers k satisfy:
k + 400 is a perfect square?
Here 1000≤k≤9999.
Optional strategy hint
Add 400 to both endpoints of the four-digit range. Find the first and last included square roots. Check the squares just outside your bounds so that no answer is missed.
values