CHAPTER 7

Question 1

ENGINEERING RANKS

The undergraduates of a School of Engineering wished to form ranks for a parade. In ranks of 3 abreast, 2 men were left over; in ranks of 5, 4 over; in 7’s, 6 over; and 11’s, 10 over. What is the least number of marchers there must have been ?

Answer 1

the least number of marchers
is 1154

Question 2

5^999,999 by 7

What is the remainder upon dividing 5^999,999 by 7

Answer 2

6

Question 3

BABY MONKEY

A pet store offered a baby monkey for sale at $1.25. The monkey grew. Next week it was offered at $1.89, then $5.13, then $5.94, then $9.18 and on the sixth week a Ph.D. in Aeronautics bought it for $12.42. How were the new prices figured?

procedure?

Answer 3

adding the square of the sum of the
digits of the previous price TO the previous price

Question 4

ARRANGE ODDS AND EVEN DIGITS

The odd digits 1, 3, 5, 7, and 9, add up to 25, while the even figures, 2, 4, 6, and 8, only add up 20. Arrange these figures so that the odd ones and the even ones add up alike. Complex and improper fractions and recurring decimals are not allowed.

Answer 4

79 + 5 1/3
84 + 2/6

both are equal to
84 1/3

Question 5

THE UNIVERSE

Assume the universe is a billion billion light years in diameter and is packed solidly with matter weighing a billion billion tons per cubic inch and each gram of this matter contains a billion billion atoms. Also, every second during the past billion billion years, a billion billion similar universes were created. Without using any symbols and restricting yourself to a total of three digits, write a number that far exceeds the total atoms of all these universes.

Answer 5

9^(9^9)

Question 6

HOW MANY MILES DROVE, ODOMETER

The sum of the digits on the odometer in my car (which reads up to 99999.9 miles) has never been higher than it is now, but it was the same 900 miles ago. How many miles must I drive before it is higher than it is now.

Answer 6

100 miles

since the odometer reading now must be x 9899.9 where x is any digit from 0 to 9.

Question 7

PRIME NUMBER OF INFINITE SERIES

How many primes are in the following infinite series where the digits are arranged in declining order? 9; 98; 987; 9876;…
987654321; 9876543219; 98165432198;… etc.

Answer 7

There are no primes in this series

Question 8

PRODUCT OF POSITIVE INTEGERS, ADD UP TO 100

What is the largest number which can be obtained as the product of positive integers which add up to 100?

• largest number (equation form)?
• add up to 100?

Answer 8

The largest possible number is 3^32 ⋅ 2^2

3(32) + 2(2) = 100

Question 9

MARTIAN FINGERS

The first expedition to Mars found only the ruins of a civilization. The explorers were able to translate a Martian equation as follows: 5x^2 - 50x + 125 = 0 x = { 5, 8. This was strange mathematics. The value x = 5 seemed legitimate enough but x= 8 required some explanation. If the Martian number system developed in amanner similar to ours, how many fingers would you say the Martians had?

Answer 9

Thus b = 5 + 8 = 13 and
the Martians had 13 fingers.

Question 10

RECTANGULAR PICTURE, WITH 1 IN. FRAME

A rectangular picture, each of whose dimensions is an integral number of inches, has an ordinary rectangular frame 1 inch wide. Find the dimensions of the picture if the area of the picture and the area of the frame are equal.

2 solutions?

Answer 10

2 integral solutions:
3 X 10
4 X 6

Question 11

UNEQUAL RATIONAL NUMBERS

Find unequal rational numbers, a, b, (other that 2 and 4) such a^b = b^a

Answer 11

a = 9/4
b = 27/8

Question 12

FIVE DIGIT NUMBER, FIRST TWO, CENTRAL, LAST TWO PERFECT SQUARE

Find a five digit number whose first two digits, central digit, and last two digits are perfect squares and whose square root is a prime palindrome.

Answer 12

36481 = 191^2

Question 13

HOUSE NUMBER AND NUMBER OF HOUSES

My house is on a road where the numbers run 1, 2, 3, 4,… consecutively. My number is a three digit one and, by a curious coincidence, the sum of all house numbers less than mine is the same as the sum of all house numbers greater than mine. What is my number and how many houses are there on my road.

Answer 13

My house number is 204, and there are 288 houses in my road

Question 14

SUM AND DIFFERENCE OF TWO PRIMES BE SQUARE

The sum and difference of two squares may be primes :
4 - 1 = 3 and 4 + 1 = 5;
9 - 4 = 5 and 9 + 4 = 13, etc.

Can the sum and difference of two primes be squares? If so, for how many different primes is this possible?

Answer 14

p = 2, q = 2

Question 15

FIRST DAY OF THE CENTURY

On what days of the week can the first day of a century fall? (The first day of the twentieth century was Jan. 1, 1901)

Answer 15

Monday

Question 16

799^3 = A^2 - B^2

Solve for A and B, both triangular numbers: 799^3 = A^2 - B^2

Answer 16

A = 800C2 = 319600
B = 799C2 = 318801

Question 17

NUMBER SQUARE IN SCALE OF 5 AND 10

A certain 6-digit number is a square in both the scale of 5 and the scale of 10. What is it?

Answer 17

232324

= 332^2 in the scale of 5
= 482^2 in the scale of 10

Question 18

STARTING WITH ONE, PLACE EACH SUCCEEDING INTEGER

Starting with one, place each succeeding integer in one of two groups such that neither group contains three integers in arithmetic progression. How far can you get?

Answer 18

first eight integers can be partitioned into:
1 2 5 6 - 3 4 7 8
1 3 6 8 - 2 4 5 7
1 4 5 8 - 2 3 6 7

9 cannot be added to any of these groups without forming an arithmetic progression.

Question 19

PAID OUT PRIZES $11 viz., $1, $11, $121

In a lottery the total prize money available was a million dollars, paid out in prizes which were powers of $11 viz., $1, $11, $121, etc. No more than 6 people received the same prize. How many prize winners were there, and how was the money distributed?

Answer 19

20 winners

Question 20

BASE NUMBER OF PUEVIGI

In the arithmetic of Puevigi, 14 is a factor of 41. What is the base of the number system?

Answer 20

B = 11. (The Puevigians use 11 as their base, since they have 5 fingers on one hand and 6 on the other.)

Question 21

n IS FOR k!

For what n is (k=1 to n)∑ k! a square?

Answer 21

Only for n = 1 or 3

Question 22

FIVE DIGITS NUMBER, IS A FACTOR OF ITS REVERSAL

Find the only number consisting of five different digits which is a factor of its reversal.

Answer 22

87912

= 4 X 21978

Question 23

NO FACTORIAL CAN END IN FIVE ZEROS

No factorial can end in five zeros. What is the next smallest number of zeros in which a factorial can not end?

Answer 23

11

Question 24

INTEGER IS A PERFECT SQUARE, CUBE AND FIFTH

One is the smallest integer which is simultaneously a perfect square, cube, and fifth power. What is the next smallest integer with this property?

Answer 24

2^30 = 1,073,741,824
perfect 30th power

try square root, cube root and fifth root, its perfect!

Question 25

BARNIE BOOKWORM, MISSING PAGES

Barnie Bookworm bought a thriller - found to his dismay,
Just before the denouement a lascicle astray.
Instead of counting one through ten, a standard cure for rages.
He totalled up the number of the missing sheat of pages.
The total was eight thousand and six hundred fifty-six.
What were the missing pages?
Try to find them just for kicks.

pages missing?
first page?
quantity of pages missing?

Answer 25

pp 255-286 are missing

p = 32 (no. of missing pages)
n = 255 (first missing page number)

Question 26

RECIPROCAL OF THE DIVISORS OF SIX SUM TO 2

The reciprocals of the divisors of six sum to two, i.e., 1/1 + 1/2 + 1/3 + 1/6 = 2. Find another number with this property.

Answer 26

28 and 496

Question 27

THE SULTAN, HOW MANY WIVES

The Sultan arranged his wives in order of increasing seniority and presented each with a golden ring. Next, every 3rd wife, starting with the 2nd, was given a 2nd ring; of these every 3rd one starting with the 2nd received a 3rd ring, etc. His first and most cherished wife was the only one to receive 10 rings. How many wives had the Sultan?

Answer 27

9,842 wives,
roughly 1 a day for 27 years

Question 28

WATER - HEAT= ICE

If you solve the alphametic WATER - HEAT= ICE, you will have the solution to this double riddle: “This bird’s assured of his breakfast/ and these before steeds cause a wreck fast” Curiously, 70243 is the answer to both riddles!

Answer 28

EARLY and CARTS
70243

W = 1, A = 0, and H = 9
10512 - 9705 = 867
10482 - 9804 = 678

Question 29

ALPHAMETIC ROMAN NUMERALS

The above alphametic involving Roman numerals is correct. It will still be correct if the proper Arabic numerals are substituted. Each letter denotes the same digit throughout and no 2 letters stand for the same digit. Find the unique solution.

Answer 29

453 + 485 = 938

Question 30

1 TO 7, FIRST AND THIRD ROW, SUM PERFECT SQUARE

Find a permutation of the numbers one through seven with the property that when placed in both the first and third rows, the seven row totals will all be perfect squares.

Answer 30

4736251

Question 31

COMPLETE THE FACTORIZATION, THEN QUIT

Using a desk calculator, a student was asked to obtain the complete factorization of 24,949,501. Dividing by successively increasing primes, he found the smallest prime divisor to be 499 with quotient 49,999. At this point, he quit. Why didn’t he carry the factorization to completion?

Answer 31

He did. 49999 is a prime and factorization is complete

Question 32

CAPITAL CONSIST OF STRAIGHT LINE SEGMENTS

Among those numbers whose literal representations in capitals consist of straight line segments only (e.g. FIVE), only one is “orthonymic”, i.e., is equal to the number of segments which comprise it. Find the number.

Answer 32

The only orthonym in English is TWENTY NINE.

Question 33

LARGE NUMBER = 13!

The numbers
6,227,020,800
6,227,028,000
6,227,280,000
are all large and roughly in the same ball park. But only one is equal to 13! Find it without use of tables, desk calculators, or hard work.

Answer 33

first number

Question 34

INSCRIPTION ON THE PURPLE MONN BOULDER

It is rumored that the above inscription appears on the purple moon boulder, a fragment of which was brought home by our Apollo 11 astronauts. If the visitors who inscribed it were humanoid, and if the plausible inference is made that it represents an addition in a place notation system, can one make a further inference as to the number of fingers these visitors had?

Answer 34

reasonable inference that the visitors had three fingers on each hand. (BASE 3)

〇 = 1
▢ = 4
△ = 3
▽ = 0

Question 35

MAGIC SQUARE, WHAT IS THE MIDDLE NUMBER

A certain magic square contains nine consecutive 2-digit numbers. The sum of the numbers in any line is equal to one of the numbers in the square with the digits reversed. This is still the case if 7 is added to each entry. What is the number in the center square?

Answer 35

17

20 | 13 | 18
15 | 17 | 19
16 | 21 | 14