14.4 Sorted numbers with sorted squares 421
Exercise
1.Show that(^16) n
(^6) n 4
=
1
4
,
(^19) n
(^9) n 5
=
1
5
,
(^26) n
(^6) n 5
=
2
5
,
(^49) n
(^9) n 8
=
4
8
.
2.Show that(16n7)^2 =27n (^8) n+1 9.
3.(3n4)^2 =1n+1 (^5) n 6.
4.John shook its head. “Multiply that huge number by 8 in my head?
You’ve got to be kidding.”
“But it’s easy, Dad.” Mike told him. “You just shift its last digit to
the front.”
The boy was right, and it is the smallest number for which it works.
What was the number?
5.John looked over his son’s shoulder. “That must be an interesting
number,” he said. “Homework?”
“Just fun, Dad,” Doug replied. “It’s the serial number on that clock
you brought back from Kaloat, and I’ve just noticed something spe-
cial about it. If you take the last two digits and put them in front,
you get exactly four times the original number, and it’s the smallest
number that works that way.”
What was the serial number?
6.Given an integern. Show that an integer can always be found which
contains only digits 0 and 1 (in the decimal scale) and which is
divisible byn.^2
7.Determine ann-digit number such that the number formed by re-
versing the digits is nine times the original number. What other
numbers besides nine are possible?
8.Write (^529) n− 2893 n− 19 as a sum of three squares of natural numbers.
9.There are only two repdigitsanwhose squares have digital sum 37.
What are these?^3
(^2) AMM Problem 4281.
(^3) Answer:.^47 and^42