A decade is defined as a complete set of consecutive nonnegative integ

Question

A decade is defined as a complete set of consecutive nonnegative integers that have identical digits in identical places, except for their units digits, with the first decade consisting of the smallest integers that meet the criteria, the second decade consisting of the next smallest integers, etc. A decade in which the prime numbers contain the same set of units digits as do the prime numbers in the second decade is the

A. fifth
B. seventh
C. eighth
D. ninth
E. eleventh

Kudos for a correct solution.

A. fifth
B. seventh
C. eighth
D. ninth
E. eleventh

AVRonaldo · Accepted Answer

A decade is defined as a complete set of consecutive nonnegative integers that have identical digits in identical places, except for their units digits

Understanding the pattern here is key ! Let's try a few examples for decades as per above definition

100, 101,102,103,104,105 ....,109 - units digit differ, all other digits are identical
200, 201,202,203,204,205.....,209
420,421,422,423,424,425......,429

From question stem, we know that the first decade consists the smallest integers that meet the above definition
1st decade - 00,01,02,03,04,05,06,07,08,09

The second decade has the next smallest integers that meet the criteria
2nd decade - 10,11,12,13,14,15,16,17,18,19

We need to find the decade in which the prime numbers contain the same set of units digits as do the prime numbers in the second decade
2nd decade - 10,11,12,13,14,15,16,17,18,19

3rd decade - 20,21 --> 21 is not prime . So stop and let's move to the next decade
4th decade - 30,31,32,33 --> 33 is not prime. Stop !
5th decade - 40,41,42,43,44,45,46,47,48,49 -- > 49 is not prime
6th decade - 50,51 --> 51 is not prime . Stop !
7th decade - 60,61,62,63 --> 63 is not prime. Stop!
8th decade - 70,71,72,73,74,75,76,77 --> 77 is not prime. Stop!
9th decade - 80,81 --> 81 is not prime. Stop!
All options have been eliminated except one. So Answer should be (E)

10th decade - 90,91,92,93 -->93 is not prime. Stop!
11th decade - 100,101,102,103,104,105,106,107,108,109 - Match !

Therefore, Correct Answer is (E) i.e eleventh decade

P.S . Took me almost 5 min to figure out the decade definition! I will definitely skip such a question in GMAT exam. :roll:

Zhenek · Answer

The 2nd decade: 10, 11, 12 ... 19. We got 4 primes there and our resulting set is (1,3,7,9).
Quick glance at the answers
A - 51 is not a prime
B - 77 is not a prime
C - 81 is not a prime
D - 93 is not a prime
E ends up being an answer

Spent like 5minutes trying to understand what was going on and then 30 seconds to actually solve the task.

sudh · Answer

a decade = 10 years

by the definition above a decade starts with a nonnegative integer (0) that have identical digits in identical places, except for their units digits
so first decade contains the years - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
second decade contains the years - 10,  11 , 12,  13 ,1 4, 15,  17 , 18,  19

we have to find next decade which has prime numbers contain the same set of units digits as the second decade
prime numbers can be expressed as (6n+1) or (6n-1)
prime numbers in the second decade

(6n-1)-type 
11 = (2-1) = 6*2 - 1
17 = (8-1) = 6*3 - 1
 (6n+1)-type 
13 = (2+1) = 6*2 + 1
19 = (8+1) = 6*3 + 1
6n-1 will yield 1 or 3 in unit's place if n = 2, 3, .. 7, 8, ... 12, 13, ... 17, 18, ... 22, 23, ...
6n+1 will yield 7 or 9 in unit's place if n = 2, 3, .. 7, 8, ... 12, 13, ... 17, 18, ... 22, 23, ...

with multiples of 7 and 8 we have 41, 43, 47, 49 (not a prime)
with multiples of 12 and 13 we have 71, 73, 77 (not prime), 79
with multiples of 17 and 18 we have 101, 103, 107, 109; matched the second decade
so the decade {100, ... 109 } is (10 -  0  + 1) 11th decade; first decade has no tens' digit

Answer E

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION:

This problem is all about reading. “A complete set of consecutive nonnegative integers…” should conjure in your mind the numbers {0, 1, 2, 3, …} – and you’re going to take a consecutive subset of those numbers. Here are the key words: “… that have identical digits in identical places, except for their units digits…” So the units digits of the numbers in a “decade” can differ, but all the other digits are the same. For example, the integers 50 through 59 would form a decade; 150 through 159 would form a different decade.

The first decade, we are told, consists of the smallest integers that meet the criteria. The smallest nonnegative integers are 0, 1, 2, 3, … and in fact, the integers 0 through 9 meet the criteria (they have identical digits in identical places, except for their units digits – and since these numbers only consist of units digits, they have no digits in common, but they’re still part of the same decade). So the first decade is {0, 1, 2, 3, …, 9}. The second decade is {10, 11, 12, 13, …, 19}, and so on.

The prime numbers in the second decade are 11, 13, 17, and 19. So you must find a decade in which the primes are xxx1, xxx3, xxx7, and xxx9 (where xxx represents the unknown identical digits in the decade). Search backwards from the answer choices, noting that since the first decade has no tens digit and the second decade has 1 as the tens digit, the “ordinal” number (first, second, third, etc.) of the decade is one more than the tens digit. That is to say, the fifth decade is {40, 41, … 49}.

(A) cannot be right, because 49 is not prime.
(B) cannot be right, because 63 and 69 are not prime.
(C) cannot be right, because 77 is not prime.
(D) cannot be right, because 81 is not prime.

Hence, the answer must be (E): 101, 103, 107, and 109 are all prime. If you really want to check, look for divisibility by primes up to the square root of the number in question—and since they’re all less than 121, which is 11^2, you can just check divisibility by 2, 3, 5, and 7. Numbers ending in 1, 3, 7, and 9 are not divisible by either 2 or 5, so you only really have to check 3 and 7.

The correct answer is E.

jayanthjanardhan · Answer

is there any way to solve this problem within 2 mins?..understanding the question itself took me like 4 mins!...

KarishmaB · Answer

Yes, the question has very unconventional language for something very simple. The concept involved is very basic. It does take some time to figure out what the question is asking and there is no other method to do that except read the question one statement at a time and try to figure out what it is saying. What helps is subtle clues such as the use of the term "decade" - perhaps every "decade" has 10 numbers.

ZulfiquarA · Answer

Irrespective of the popular opinion I understood the question within time, in like 1:30 mins but I chickened out with the difficulty on how long will it take and guessed the wrong answer haha

Adarsh_24 · Answer

How is 01,02,03,.. considered a decade?

nonnegative integers that have identical digits in identical places, except for their units digits

Is 01 considered an integer? I thought the set would exclude single digit numbers and go like

10,11,12,...
20,21,...
...

Or should I have thought of a decade in real life ?

KHannigan · Answer

Several question design issues.

Firstly  "complete set"  does not necessarily mean  "a set extended until the criterion is no longer true" . It could be argued that {13,14,15} is a decade.

Secondly, the definition -:
 "  that have identical digits in identical places, except for their units digits, with the first decade consisting of the smallest integers that meet the criteria"

- presupposes that the integers involved have non-unit digits to compare — i.e., that the condition of ‘identical digits in identical non-unit places’ applies. This excludes single-digit numbers, since they lack such structure. This is an example of existential import (presupposition of existence / presupposition of relevant structure): the condition implies the existence of relevant digit places, which 1–9 do not satisfy..

Based the definition given, the first decade should be  {11,12,13,14,15,16,17,18,19}

This means that the second decade is  {21,22,23,24,25,26,27,28,29}. 
The primes in this set are  23 , and  29 .

Therefore the next decade wherein the primes share the same set of units digits as the second, is the  fifth :  {51,52,53,54,55,56,57,58,59}  where the primes are  53  and  59 .

A decade is defined as a complete set of consecutive nonnegative integ

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A decade is defined as a complete set of consecutive nonnegative integ

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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Perfect 805 Score Debrief by Julia

My Rewards