Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

Determine the remainder of 194 (mod 11). Show that it is 7.

Formal statement:

theorem "194 mod 11 = (7::nat)"

Answer:Yes, the translation is faithful.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

What is the units digit of the product of all of the odd integers between 0 and 12? Show that it is 0.

Formal statement:

theorem "(0 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12) mod 10 = (0::nat)"

Answer:No, the product (0 * 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12) should only consist of odd numbers but "0" for example is an even number.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

What digit must be placed in the blank to make the four-digit integer $20\_7$ a multiple of 11? Show that it is 9.

Formal statement:

theorem "(2079::int) mod 11 = (0::int)"

Answer:No, $20\_7$ is altered to $207\_$ and thus the translation is not faithful.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

How many 3-letter sequences can be formed if the second letter must be a vowel (A, E, I, O, or U), and the third letter must be different from the first letter? Show that it is 3250.

Formal statement:

theorem fixes x :: nat assumes "x = 3250" shows "x = 3250"

Answer:No, the formal statement is irrelevant to the informal statement and we should not put what we want to show in the assumption.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

Solve for $x$: $x-2x+3x-4x+5x-6x+7x-8x+9x-10x=50$. Show that it is -10.

Formal statement:

theorem fixes x :: real assumes "x - 2 * x + 3 * x - 4 * x + 5 * x - 6 * x + 7 * x - 8 * x + 9 * x - 10 * x = 50" shows "x = -10"

Answer:Yes, the translation is faithful.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

Remmy wants to divide $10$ by $\frac{2}{3}$, but he cannot remember how to do that. By what number should he multiply $10$ to get the answer? Show that it is \frac{3}{2}.

Formal statement:

theorem fixes x :: real assumes h0 : "x = 10 / (2/3)" shows "x = 3/2 * 10"

Answer:Yes, the translation is faithful.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

Let $g(x) = x^2 - 11x + 30.$ Find the polynomial $f(x),$ with positive leading coefficient, that satisfies \[g(f(x)) = x^4 - 14x^3 + 62x^2 - 91x + 42.\] Show that it is x^2-7x+6.

Formal statement:

theorem fixes f g :: "real \<Rightarrow> real" assumes h0 : "\<And>x. f x = x^2 - 7 * x + 6" and h1 : "\<And>x. g x = x^2 - 11 * x + 30" and h2 : "\<And>x. g (f x) = x^4 - 14 * x^3 + 62 * x^2 - 91 * x + 42" shows "f = (\<lambda>x. x^2 - 7 * x + 6)"

Answer:No, what we want to show i.e. f = (\<lambda>x. x^2 - 7 * x + 6) should not be put in the assumption.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

Each of four circles has a radius of 4 inches. A larger circle has an area equal to the total area of the four smaller circles. What is the length of the radius, in inches, of the larger circle? Show that it is 6.

Formal statement:

theorem fixes x :: real assumes "x > 0" and "x^2 = 4^2 * 4" shows "x = 6"

Answer:Yes, the translation is faithful although the formal statement itself is incorrect.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

The sum of four consecutive positive even integers is a perfect square. What is the least possible sum? Show that it is 20.

Formal statement:

theorem "(2 + 4 + 6 + 8) = (20::nat)"

Answer:No, the sum needs to be a perfect square but "20" is not.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

What is the least positive integer $n$ such that $80325$ divides $n!$? Show that it is 1155.

Formal statement:

theorem "(80325::nat) dvd (fact 1155)"

Answer:No, the formal statement misses the "least" condition. That is, there could be a number $n$ < 1155 and $80325$ divides $n!$.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

Find $9^{-1} \pmod{100}$, as a residue modulo 100.  (Give an answer between 0 and 99, inclusive.) Show that it is 2.

Formal statement:

theorem "9*2 mod 100 = (1::nat)"

Answer:Yes, the translation is faithful although the formal statement itself is incorrect.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

Find the units digit of the following within the indicated number base: $(14_8)^2$ Show that it is 4.

Formal statement:

theorem "(14::nat) ^ 2 mod 8 = (4::nat)"

Answer:No, the number "14" should also be in base 8 as indicated by the question.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

What is the last digit of the base $6$ representation of the base ten integer $355$? Show that it is 1.

Formal statement:

theorem "(355::nat) mod 6 = (1::nat)"

Answer:Yes, the translation is faithful.

Is the following Isabelle formal statement a faithful translation of the informal statement, Yes or No?

Informal statement:

Convert $222_{10}$ to base $13$. Use digits A, B, and C if necessary. Show that it is 173_{13}.

Formal statement:

theorem "digits_in_base 222 13 = [3,7,1]"

Answer:Yes, the translation is faithful.