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SAT Numbers Practice Question: Factorization and Divisibility
Question 1: Numbers : Remainders and Divisibility
What is the remainder when 163 + 173 + 183 + 193 is divided by 70?
- 0
- 69
- 1
- 34
- 47
-
Correct Answer
Choice A. The remainder is 0.
Explanatory Answer
This one is a number properties question and tests your understanding of few rules of factorization of polynomials and presents it as a remainder question. The rules pertaining to factorization are given in the first tab. Using those rules, try and solve the question yourself. If you need additional help click on the next tab to get a detailed explanation to this numbers practice question.
Factorization Rules pertaining to polynomials
- \x^n−y^n = \lbrack x−y \rbrack \lbrack x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1}\rbrack \\)
It is therefore, evident that \x^n−y^n \\) is divisible by \\lbrack {x - y} \rbrack \\) when \x \ne y \\). - \x^3 + y^3 \\) = \\lbrack {x + y} \rbrack \lbrack x^2 - xy + y^2 \rbrack \\)
- In general, when n is an odd positive integer, \x^n + y^n \\) is divisible by \\lbrack {x + y} \rbrack \\), given \{x + y} \ne 0 \\)
Explanatory Answer
Let us rewrite the expression as \\lbrace 16^3 + 19^3 \rbrace + \lbrace 17^3 + 18^3 \rbrace \\)
From the rules given in the earlier tab, the expression can be written as
(16 + 19)(162 - 16 * 19 + 192) + (17 + 18)(172 - 17*18 + 182)
= 35 (162 - 16 * 19 + 192 + 172 - 17*18 + 182)
We can deduce the result as 35 (even - even + odd + odd - even + even)
35 (even)
When 35 is multiplied by an even number, the product will be a multiple of 70.
Or the expression is divisible by 70.
The remainder is 0.