third by a fourth power. 8. (Brahmagupta, 7th century A.D.) When eggs in a basket are removed 2, 3, 4, 5, 6 at a time there remain, respectively, 1, 2, 3, 4, 5 eggs. When they are taken out 7 at a time, none are left over. Find the smallest number of eggs that could have been contained in the basket. haalrot lrot of ng
third by a fourth power. 8. (Brahmagupta, 7th century A.D.) When eggs in a basket are removed 2, 3, 4, 5, 6 at a time there remain, respectively, 1, 2, 3, 4, 5 eggs. When they are taken out 7 at a time, none are left over. Find the smallest number of eggs that could have been contained in the basket. haalrot lrot of ng
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
#8
![THE THEORY OF CONGRUENCES
83
7 (a) Obtain three consecutive integers, each having a square factor.
[Hint: Find an integer a such that 22 | a, 32 | a + 1, 5² | a + 2.]
(6) Obtain three consecutive integers, the first of which is divisible by a square, the
second by a cube, and the third by a fourth power.
8. (Brahmagupta, 7th century A.D.) When eggs in a basket are removed 2, 3, 4, 5, 6 at a
time there remain, respectively, 1, 2, 3, 4, 5 eggs. When they are taken out 7 at a time,
none are left over. Find the smallest number of eggs that could have been contained in
the basket.
9. The basket-of-eggs problem is often phrased in the following form: One egg remains
when the eggs are removed from the basket 2, 3, 4, 5, or 6 at a time; but, no eggs remain
if they are removed 7 at a time. Find the smallest number of eggs that could have been
in the basket.
10. (Ancient Chinese Problem.) A band of 17 pirates stole a sack of gold coins. When they
tried to divide the fortune into equal portions, 3 coins remained. In the ensuing brawl over
who should get the extra coins, one pirate was killed. The wealth was redistributed, but
this time an equal division left 10 coins. Again an argument developed in which another
pirate was killed. But now the total fortune was evenly distributed among the survivors.
What was the least number of coins that could have been stolen?
11. Prove that the congruences
X = a (mod n)
and
X = b (mod m)
admit a simultaneous solution if and only if gcd(n, m)|a – b; if a solution exists, confirm
that it is unique modulo lcm(n, m).
12. Use Problem 11 to show that the following system does not possess a solution:
and
x = 7 (mod 15)
(9 pou) s = x
13. If x = a (mod n), prove that either x = a (mod 2n) or x = a +n (mod 2n).
14. A certain integer between 1 and 1200 leaves the remainders 1, 2, 6 when divided by 9,
11, 13, respectively. What is the integer?
15. (a) Find an integer having the remainders 1, 2, 5, 5 when divided by 2, 3, 6, 12, respec-
tively. (Yih-hing, died 717).
(b) Find an integer having the remainders 2, 3,4, 5 when divided by 3, 4, 5, 6, respectively.
(Bhaskara, born 1114).
c) Find an integer having the remainders 3, 11, 15 when divided by 10, 13, 17, respec-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F92d20c31-ffe0-433e-8ae9-f10b1ae32c10%2Fy5k82eo.jpeg&w=3840&q=75)
Transcribed Image Text:THE THEORY OF CONGRUENCES
83
7 (a) Obtain three consecutive integers, each having a square factor.
[Hint: Find an integer a such that 22 | a, 32 | a + 1, 5² | a + 2.]
(6) Obtain three consecutive integers, the first of which is divisible by a square, the
second by a cube, and the third by a fourth power.
8. (Brahmagupta, 7th century A.D.) When eggs in a basket are removed 2, 3, 4, 5, 6 at a
time there remain, respectively, 1, 2, 3, 4, 5 eggs. When they are taken out 7 at a time,
none are left over. Find the smallest number of eggs that could have been contained in
the basket.
9. The basket-of-eggs problem is often phrased in the following form: One egg remains
when the eggs are removed from the basket 2, 3, 4, 5, or 6 at a time; but, no eggs remain
if they are removed 7 at a time. Find the smallest number of eggs that could have been
in the basket.
10. (Ancient Chinese Problem.) A band of 17 pirates stole a sack of gold coins. When they
tried to divide the fortune into equal portions, 3 coins remained. In the ensuing brawl over
who should get the extra coins, one pirate was killed. The wealth was redistributed, but
this time an equal division left 10 coins. Again an argument developed in which another
pirate was killed. But now the total fortune was evenly distributed among the survivors.
What was the least number of coins that could have been stolen?
11. Prove that the congruences
X = a (mod n)
and
X = b (mod m)
admit a simultaneous solution if and only if gcd(n, m)|a – b; if a solution exists, confirm
that it is unique modulo lcm(n, m).
12. Use Problem 11 to show that the following system does not possess a solution:
and
x = 7 (mod 15)
(9 pou) s = x
13. If x = a (mod n), prove that either x = a (mod 2n) or x = a +n (mod 2n).
14. A certain integer between 1 and 1200 leaves the remainders 1, 2, 6 when divided by 9,
11, 13, respectively. What is the integer?
15. (a) Find an integer having the remainders 1, 2, 5, 5 when divided by 2, 3, 6, 12, respec-
tively. (Yih-hing, died 717).
(b) Find an integer having the remainders 2, 3,4, 5 when divided by 3, 4, 5, 6, respectively.
(Bhaskara, born 1114).
c) Find an integer having the remainders 3, 11, 15 when divided by 10, 13, 17, respec-
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