Question

Find the smallest positive integer whose cube ends in 888 (of course, do this without a calculator or computer).

Answer #1

Find the smallest positive integer n for which n2 + n
+ 17 is composite.

Find the smallest positive integer x such that: x mod 2=1 x mod
3=2 and x mod 5=3
What is the next integer with this property?

The least common multiple of nonzero integers a and b is the
smallest positive integer m such that a | m and b | m; m is usually
denoted [a,b]. Prove that
[a,b] = ab/(a,b) if a > 0 and b > 0.

What is the smallest positive integer x satisfying x ≡
1 mod 9, x ≡ 6 mod 10, x ≡ 4 mod 11?

A point mass M is suspended in the middle of a string whose ends
are fixed at the same level distance 2ℓ apart; the mass is distance
h below the ends of the string. The total mass of the string is m
< M. (a) Find the speed of sound waves along the string and (b)
the principal frequency of string vibrations on either side of the
mass. (c) Do you think this is the smallest vibration frequency in
the...

Use the Euclidean Algorithm to find the smallest nonnegative
integer x, s.t. there exists integers k_1, k_2, such that x = 11k_1
+ 3 and x = 49k_2 + 5.

Find two positive numbers whose product is 253 and whose sum is
a minimum.

1) Find two positive numbers whose sum is 31 and product is
maximum.
2) Find two positive whose product is 192 and the sum is
minimum.

Without using induction, prove that for x is an odd, positive
integer, 3x ≡−1 (mod 4). I'm not sure how to approach the problem.
I thought to assume that x=2a+1 and then show that 3^x +1 is
divisible by 4 and thus congruent to 3x=-1(mod4) but I'm stuck.

find the least positive residue modulo 51 of the following integer
2^1282.

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