22.7Chapter 22 Exercise
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Chapter 22 · Consecutive Natural Numbers

Lesson 22.7 — Chapter 22 Exercise

A complete English student edition of Test 22. Work with arithmetic sequences, digit counts, consecutive composite, prime, even, and odd numbers, range averages, concatenated numerals, carry structure, and complete justifications.

14 questions116 pointsoriginalAutosave + print

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Tips for this exercise

Sequence vocabularyConsecutive natural numbers differ by 1; consecutive even or odd numbers differ by 2. Consecutive primes are neighboring entries in the ordered prime list, so their gaps need not be equal.
Range and numeral conventionsAll stated endpoints are included. A concatenation such as 1234567891011… is written with no spaces or separators. On this page, a natural number means a positive integer.
Read the expression carefullyRead the clarification beside Question 5 before answering. Questions 11–14 also require written explanations.

Section I · Fill-in questions

Questions 1–10 · 6 points each · 60 points

60 points
1

Sum a ten-term arithmetic sequence

Not answered

Evaluate:

99 + 198 + 297 + 396 + 495 + 594 + 693 + 792 + 891 + 990
Optional arithmetic-sequence work
2

Add all two-digit positive integers

Not answered

Find the sum of every two-digit positive integer:

10 + 11 + 12 + ··· + 98 + 99.
Optional first–last pairing work
3

Count numbers containing the digit 5

Not answered

Among the integers from 1 through 100, inclusive, how many contain the decimal digit 5 at least once?

integers
Optional complement or case-count work
4

Find the first run of five composite numbers

Not answered

Five consecutive natural numbers are all composite. What is the least possible value of their sum?

Consecutive
The numbers have the form n,n+1,n+2,n+3,n+4.
Composite
Each number is greater than 1 and has more than two positive divisors.
Optional search and minimality proof

Give a run that works, then explain why no earlier run of five integers can work. A prime interrupts a run of composite numbers.

5

Evaluate the expression exactly as printed

Not answered

Evaluate:

1 + 2 + 3 + 5 + 6 + ··· + 48 + 49 + 50 + 49 + 48 + 47 + ··· + 7 + 6 + 5
original integrity noteThe rising side in the worked example jumps directly from 3 to 5; the term 4 is omitted at that point. This worksheet asks you to evaluate the expression exactly as shown. For this question, keep the skip: do not add 4. The descending side stops at 5, exactly as shown.
Optional grouping or pairing work
6

Use parity in a block of seven consecutive primes

Not answered

Seven consecutive primes are written from greatest to least as:

a, b, c, d, e, f, g.

Their total is even. Find d.

Consecutive primesThese are neighboring entries in the ordered list of primes. Their numerical gaps are not necessarily equal.
Optional parity reasoning
7

Relate the extremes of seven consecutive even numbers

Not answered

Seven consecutive positive even numbers are arranged in increasing order. The greatest is exactly three times the least. Find the greatest number.

n, n+2, n+4, n+6, n+8, n+10, n+12
Optional equation work
8

Find the least three-odd-number total meeting a bound

Not answered

Three consecutive positive odd numbers have a sum of at least 70. What is the least possible value of the smallest odd number?

Optional boundary and previous-case check
9

Average all multiples of 3 from 1 through 100

Not answered

Among the integers from 1 through 100, find the average of all numbers divisible by 3.

Optional first–last pairing work
10

Count the digits in a concatenated numeral

Not answered

Write the natural numbers from 1 through 35 consecutively with no spaces or separators:

1234567891011···333435

How many decimal digits does the resulting numeral contain?

digits
Optional one-digit and two-digit block work

Section II · Extended-response questions

Questions 11–14 · 14 points each · 56 points

56 points
11

Sum the integers that are not divisible by 9

Not answered

Among the positive integers from 1 through 100, find the sum of all integers that are not divisible by 9.

12

Recover the greatest of twenty consecutive positive integers

Not answered

Twenty consecutive positive integers have total:

290.

Find the greatest integer.

13

Count two-digit numbers equal to four times their digit sum

Not answered

A two-digit positive integer is four times the sum of its two decimal digits. How many such two-digit integers are there?

If the numeral is ab, then 10a+b = 4(a+b).

Here a is the tens digit (1–9), and b is the units digit (0–9). Both are whole numbers.

numbers
14

Track digit sums through three carries

Not answered

Positive integers A and B satisfy:

  • The sum of the decimal digits of A is 35.
  • The sum of the decimal digits of B is 26.
  • The usual decimal addition A+B produces exactly three carries.

Find the sum of the decimal digits of A+B.

Carry conventionCount a carry whenever one place sends 1 into the next place in the standard addition algorithm.