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# Lowering which of the following numbers by one will result in the leas

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Math Expert
Joined: 02 Sep 2009
Posts: 55230
Lowering which of the following numbers by one will result in the leas  [#permalink]

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20 Jul 2017, 21:56
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Difficulty:

65% (hard)

Question Stats:

36% (01:10) correct 64% (01:09) wrong based on 52 sessions

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$$(17 * 19 * 23 * 29)^{31} = n$$
Lowering which of the following numbers by one will result in the least decrease of n?

A. 17
B. 19
C. 23
D. 29
E. 31

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Lowering which of the following numbers by one will result in the leas  [#permalink]

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21 Jul 2017, 11:31
4
Bunuel wrote:
$$(17 * 19 * 23 * 29)^{31} = n$$
Lowering which of the following numbers by one will result in the least decrease of n?

A. 17
B. 19
C. 23
D. 29
E. 31

When faced with huge numbers and/or exponents, use smaller numbers and/or exponents that follow the prompt's pattern.

If we are looking for the least change (decrease), we need the value that has the most weight to be decreased a little.

To find the smallest decrease in $$n$$, we need the value that has the most weight to be decreased a little.
• Changing a small number such as 2, to 1, cuts the product in half.
• Changing a larger number such as 4, to 3, decreases the product by 1/4.

(And that which gets decreased must be a factor, not the exponent. Decreasing the exponent by 1, see below, is not decreasing by a little.)

Think of n = (2*7)$$^2$$. That equals 196.

By definition, an exponent yields exponential increase in value.
Decreasing the exponent will yield an exponential decrease - not likely to be the least change.

(2*7)$$^2$$ = 196

Decrease exponent 2 by 1. (2*7)$$^1$$ = 14

Now decrease factor 2 by 1. (1*7)$$^2$$ = 49

Now decrease factor 7 by 1. (2*6)$$^2$$ = 144

Original value => 196
Exponent: 2 - 1 => 14
Factor 2 - 1 => 49
Factor 7 - 1 => 144

The smallest decrease comes from decreasing the largest factor by 1.

You can use (2*3*4*5)$$^3$$ (increasing positive factors to an odd power) and come up with the same result. The essential "ingredients" in the prompt's $$n$$ are the same.
When I see insane values, I think: go with small but similar values.

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Lowering which of the following numbers by one will result in the leas   [#permalink] 21 Jul 2017, 11:31
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