ABACUS International Math Challenge
for
7th and 8th graders
December, 1998
C.89. How many times does the digit 1 appear
in base 10 in the number N=9+99+999+...+999...99, if the last addend has
1998 digits?
C.90. Find the smallest positive whole
number that defeats the following statement:
"If the sum of the digits of a whole number n is divisible by 6,
then n is divisible by 6."
C.91. Write a digit in front and at the
end of the number 1998, so that the new 6-digit number is divisible by 99.
C.92. Write a digit in front and at the
end of the number 1998, so that the new 6-digit number is divisible by 88.
C.93. Find the smallest positive whole
number in which the number created from the first two digits is divisible
by 2, the number created from the first three digits is divisible by 3,
..., the number created from the first eight digits is divisible by 8, and
the number itself is divisible by 9.
C.94. Using the digits 1, 2, 3, 4, 5, and
6 once and only once, find 6-digit numbers in which the number created from
the first two digits is divisible by 2, the number created from the first
three digits is divisible by 3, the number created from the first four digits
is divisible by 4, the number created from the first five digits is divisible
by 5, and the number itself is divisible by 6.
C.95. Using the digits 1, 2, 3, 4, 5, 6,
7, 8, and 9 once and only once, find 9-digit numbers in which the number
created from the first two digits is divisible by 2, the number created
from the first three digits is divisible by 3, ..., the number created from
the first eight digits is divisible by 8, and the number itself is divisible
by 9.
C.96. By adding an extra digit anywhere
in the number 975 312 468, create a 10-digit number that is divisible by
33.
Please, send your solutions to: